Non-conforming high order approximations of the elastodynamics equation

Comput. Methods Appl. Mech. Engrg. 209–212 (2012) 212–238

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Comput. Methods Appl. Mech. Engrg.

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Review

Non-conforming high order approximations of the elastodynamics equation

P.F. Antonietti a, I. Mazzieri a,⇑, A. Quarteroni a,c, F. Rapetti b

a MOX-Modelling and Scientific Computing, Department of Mathematics, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133, Milano, Italyb Université de Nice Sophia Antipolis, Laboratoire de Mathématiques J.A. Dieudonné, Parc Valrose, 06108 Nice, Cedex 02, Francec CMCS-MATHICSE, École Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 March 2011Received in revised form 24 September 2011Accepted 4 November 2011Available online 15 November 2011

Keywords:Spectral methodsNon-conforming domain decompositiontechniquesComputational seismologyNumerical approximations and analysis

1

0045-7825/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cma.2011.11.004

⇑ Corresponding author. Tel.: +39 02 2399 4604.E-mail addresses: [email protected] (P.F.

(F. Rapetti).

In this paper we formulate and analyze two non-conforming high order strategies for the approximationof elastic wave problems in heterogeneous media, namely the Mortar Spectral Element Method and theDiscontinuous Galerkin Spectral Element Method. Starting from a common variational formulation wemake a full comparison of the two techniques from the points of view of accuracy, convergence, grid dis-persion and stability.

� 2011 Elsevier B.V. All rights reserved.

Contents

1. Introduction and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2132. Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2133. Non-conforming Galerkin spectral formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

3.1. Discontinuous Galerkin spectral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2163.2. Mortar spectral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

4. Algebraic formulations and time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.1. Algebraic formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 4.2. Structural damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.3. Time integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5. Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5.1. Semi-discrete error estimates-DGSEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2205.2. Semi-discrete error estimates-MSEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.3. Fully-discrete error estimates – DGSEM/MSEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6. Analysis of grid dispersion and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.1. Grid dispersion – DGSEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2226.2. Grid dispersion – MSEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2236.3. Grid dispersion – numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2236.4. Stability – DGSEM and MSEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266.5. Stability – numerical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
7. Accuracy and order of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2288. An application of geophysical interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2319. Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

9.1. Semi-discrete error analysis – DGSEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2349.2. Semi-discrete error analysis – MSEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

0. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

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Antonietti), [email protected] (I. Mazzieri), [email protected] (A. Quarteroni), [email protected]

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P.F. Antonietti et al. / Comput. Methods Appl. Mech. Engrg. 209–212 (2012) 212–238 213

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

1. Introduction and motivations

The possibility of inferring the physical parameter distributionof the Earth’s substratum, from information provided by elasticwave propagations, has increased the interest towards computa-tional seismology. Recent developments in this scientific disciplineconcern with different numerical strategies as finite differences, fi-nite elements, but the major efforts apply to spectral elementmethods (see [1–7]).

A motivation is that, in geophysical or industrial applications, fi-nite difference discretizations require very large systems of equa-tions to model realistic rock properties and uniform meshes areneeded. On the other hand, when classical finite element methodsare employed for treating complex geometries, it is necessary to in-vert the mass matrix.

The reasons for using spectral element-based approximationscan be summarized in the following lines. Firstly, the flexibilityin handling complex geometries, retaining the spatial exponentialconvergence for locally smooth solutions. Secondly, since spectralelement methods are based on the weak formulation of the elasto-dynamics equations, they handle naturally both interface continu-ity and free boundary conditions, allowing very accurateresolutions of evanescent interface and surface waves (of majorinterest in seismology). Finally, spectral element methods retaina high level parallel structure, thus well suited for parallelcomputers.

However, when dealing with complex wave phenomena, suchas soil–structure interaction problems or seismic response of sed-imentary basins, the geometrical and polynomial flexibility is animportant task for simulating correctly the wave-front field.

For this reason we consider two different non-conforming high-order techniques, namely the Mortar Spectral Element Method(MSEM) [8,9] and the Discontinuous Galerkin Spectral ElementMethod (DGSEM) [10–12] to simulate seismic wave propagationin heterogeneous media. In contrast to standard conforming dis-cretizations, as Spectral Element Method (SEM) [13,14], these tech-niques have the further advantages that they can accommodatediscontinuities, not only in the parameters, but also in the wave-field, while preserving the energy.

Depending on the involved materials it is possible to make apartition of the computational domain. Then, in each non-overlap-ping subregion a spectral finite element discretization is employed.The quadrilaterals/hexahedras do not have to match betweenneighbouring subdomains, and different spectral approximationdegrees are allowed. Therefore, the continuity of the solution atthe skeleton of the decomposition is imposed weakly, either bymeans of a Lagrange multiplier for the MSEM, or by penalizingthe jumps of the displacement on the skeleton in the DGSEM.

In the present work, starting from a displacement-based weakformulation of the elastodynamics equation, we analyze stability,convergence, accuracy, dissipation and dispersion for the MSEMand DGSEM for the space discretization combined with second or-der time integration scheme. In particular we prove a priori errorbounds for both the semi-discrete and fully-discrete non-conform-ing methods.

A similar analysis is provided in the existing literature for aslightly different Discontinuous Galerkin formulation, for dynamiclinear elasticity and viscoelasticity [12,15]. In fact the above formu-lation involves an additional penalty term whose physical meaningis unclear. Yet, other authors refer to that analysis when discussing

their Discontinuous Galerkin schemes [16,17]. Here we modify andupdate the results of [12] to analyze the presented DGSEM.

In the MSEM case, at the best of our knowledge, such analysishas never been carried out before in elastodynamics, but only forelliptic and parabolic equations [8,18–20].

Since we are dealing with time-dependent problems, we alsotake into account of the stability and dispersion property of ournumerical scheme.

For wave propagation problems, the grid dispersion criteriondetermines the lowest number of nodes per wavelength such thatthe numerical solution has an acceptable level of accuracy, whilethe stability criterion determines the largest time step allowedfor explicit time integration schemes.

A general framework to study the numerical dispersion for theSEM was developed in [21] and analyzed for the acoustic case up topolynomial approximation degree equal to three. In [22] a com-plete description for the elastic case is given, based on a Rayleighquotient approximation of the eigenvalue problem characterizingthe dispersion relation.

For the DGSEM, grid dispersion has been analyzed in [23,16]. Inparticular in [23] the dispersion and dissipation errors of theacoustic wave equation in one space dimension are derived usingthe flux formulation. The results include polynomial approxima-tion degree equal to three and conjectures on the extension tohigher degrees are given. Making use of the plane wave analysis,in [16] a complete description of the grid dispersion properties iscarried out for both the acoustic and the elastic case.

At the best of our knowledge, for the MSEM no results are avail-able for the grid dispersion properties regarding the elastic waveequation.

For what concerns the stability, a classical numerical approachto solve a second order initial value problem is provided by thefamily of the Newmark methods [24]. The Leap-Frog Finite Differ-ence Method is a special case of that family which is second orderaccurate, explicit and conditionally stable, and is the most popularone used in seismic modelling [4,21,25–27]. Other schemes likeRunge–Kutta or Taylor–Galerkin, are used too [17,3,7].

In this work we derive, for the Leap-Frog Method, stabilitybounds linking the time step with the size of the elements andthe maximum wave velocity. All results obtained are comparedto those obtained with the conforming SEM case.

After introducing the elastodynamics problem and its varia-tional formulation in Section 2, we describe in Section 3 the geo-metrical and functional discretization of the problem within thecontext of non-conforming approximations. In particular we derivethe Mortar and the Discontinuous Galerkin Spectral Formulations.The algebraic aspects of the two methods are then described inSection 4. Section 5 is focused on the convergence estimates whileSection 6 is devoted to the grid dispersion and stability analysis,which are carried out for 2-d case. In Sections 7 and 8 we discussthe property of accuracy and convergence of the MSEM and theDGSEM, and present a geophysical application, namely the seismicresponse of an alluvial basin, respectively. Finally in Section 9 wereport the proofs of the convergence estimates given in Section 5.

2. Problem formulation

Let us consider an elastic medium occupying a finite regionX � Rd; d ¼ 2;3, with boundary C = oX and unit outward normal

214 P.F. Antonietti et al. / Comput. Methods Appl. Mech. Engrg. 209–212 (2012) 212–238

n. The boundary is assumed to be composed of portions CD, wherethe displacement vector u is prescribed, CN where external loadsapply, and CNR where suitable non-reflecting conditions are im-posed. The portion CNR is in fact a fictitious boundary of the com-putational domain which is introduced to bound the physicaldomain for the numerical approximation of wave propagationproblems in unbounded media. We make the assumptions thateither CD or CN can be empty, CD \ CN = ; and CN \ CNR = ;.

Here and in the sequel, an underlying bar denotes matrix or ten-sor quantities, while vectors are typed in bold. Having fixed thetemporal interval [0,T], with T real and positive, the equilibriumequations for an elastic medium, subjected to an external force fread:

q@ttu�r � rðuÞ ¼ f; in X� ½0; T�;u ¼ 0; on CD � ½0; T�;rðuÞ � n ¼ t; on CN � ½0; T�;non reflecting boundary conditions on CNR � ½0; T�;@tu ¼ u1; in X� f0g;u ¼ u0; in X� f0g;

8>>>>>>>><>>>>>>>>:ð1Þ

where u is the medium displacement vector, r the stress tensor, tthe time variable and q the material density. Without loss of gener-ality (see, for instance, [28]) we make the following further assump-tions on C: on CD the medium is rigidly fixed in the space and on CN

we prescribe surface tractions t. Finally, on CNR non-reflectingboundary conditions are imposed: from the mathematical point ofview, the latter have the effect of introducing a fictitious tractiont⁄ which is a linear combination of space and time derivatives ofthe displacement u (cf. [29,27], for example). In particular ford = 2, if CNR has outward unit normal n = (nx,ny) and tangential unitvector s = (sx,sy), the non-reflecting conditions in coordinate frame{s,n} take the form

@@n ðu � nÞ ¼ � 1

cP

@@t ðu � nÞ þ

cS�cPcP

@@s ðu � sÞ;

@@n ðu � sÞ ¼ � 1

cS

@@t ðu � sÞ þ

cS�cPcP

@@s ðu � nÞ:

(ð2Þ

For d = 3 non-reflecting boundary conditions are given by

@@n ðu � nÞ ¼ � 1

cP

@@t ðu � nÞ þ

cS�cPcP

@@s1ðu � s1Þ þ @

@s2ðu � s2Þ

h i;

@@n ðu � s1Þ ¼ � 1

cS

@@t ðu � s1Þ þ cS�cP

cP

@@s1ðu � nÞ;

@@n ðu � s2Þ ¼ � 1

cS

@@t ðu � s2Þ þ cS�cP

cP

@@s2ðu � nÞ;

8>>><>>>: ð3Þ

where s1 and s2 are two arbitrary mutually orthogonal unit vectorson the plane orthogonal to n, the normal to CNR, such that {s1,s2,n}defines a right handed Cartesian frame.

The quantities cP and cS appearing in (2) and (3) are respectivelythe compressional and the shear wave velocities, defined as

cP ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikþ 2l

q

sand cS ¼

ffiffiffiffilq

r; ð4Þ

where k and l are the Lamé elastic coefficients. We remark that forheterogeneous media q, k and l are bounded functions of the spa-tial variable, not necessarily continuous, i.e., q, k and l 2 L1(X). Weobserve that Neumann type boundary conditions can be simply gi-ven by (2) or (3), where the right-hand side is substituted with theknown value of the external load t.

To complete the system in (1), we prescribe initial conditions u0

and u1 for the displacement and the velocity, respectively. Whenwe consider viscoelastic materials, see Section 7, we introduce inthe system (1) an additional term in the form of volume forcesfvisc ¼ �2qf _u� qf2u, where f is a suitable decay factor withdimension inverse of time. Correspondingly, the equation of mo-tion becomes

q@ttu�r � rðuÞ ¼ f þ fvisc: ð5Þ

The parameter f is spatially variable (i.e. piecewise constant), as in[27], in order to model absorbing regions, thus providing an alterna-tive or a complement to the absorbing boundary conditions. Inother cases, like seismic wave propagation through heterogeneousmedia with strong elastic impedance, this model is used to preventthe onset of non-physical oscillations of the numerical solution.

We consider the strain tensor e defined as the symmetric gradi-ent of u, i.e.,

eðuÞ ¼ 12ruþru>� �

;

so that the stress tensor r satisfies the constitutive relation(Hooke’s law)

rðuÞ ¼ kr � uIþ 2leðuÞ ¼ DeðuÞ;

where I is the d-dimensional identity tensor and D is the fourth or-der positive definite Hooke’s tensor, satisfying the symmetries

Dijk‘ ¼ Djik‘ ¼ Dij‘k ¼ Dk‘ij:

Here and in the sequel we use the standard notation [30] to definethe L2-inner product (�, �)X for scalar, vector and tensor quantities.

By multiplying the first equation in (1) for a regular enoughfunction v (candidate to represent an admissible displacement),integrating by parts over the domain X, using the Green’s formula:

� r � rðuÞ;vð ÞX ¼ rðuÞ; eðvÞð ÞX � v;rðuÞ � nð ÞC;

and imposing the boundary conditions, the variational formulationof (1) reads: "t 2 (0,T] find u = u(t) 2 V such that

dttðqu;vÞX þAðu;vÞX ¼ LðvÞ 8v 2 V ; ð6Þ

where the bilinear form A : V � V ! Rd is defined as

Aðu;vÞX ¼ rðuÞ; eðvÞð ÞX;

and the linear functional L : V ! Rd as

LðvÞ ¼ t;vð ÞCNþ t�;vð ÞCNR

þ f;vð ÞX:

Here V is the Sobolev space V = {v 2 [H1(X)]d :v = 0 on CD}, whereL2(X) is the space of square integrable functions over X andH1(X) is the space of functions in L2(X) with gradient in [L2(X)]d.We recall that the bilinear form Að�; �Þ is symmetric, V-elliptic andcontinuous [31]. These conditions imply that problem (6) admitsa unique solution u 2 C0ðð0; TÞ; VÞ \ C1ðð0; TÞ; ½L2ðXÞ�dÞ satisfying sta-bility estimates [32,31], provided that q 2 L1(X) is a strictly posi-tive function, and that u0 2 V, u1 2 [L2(X)]d and f 2 [L2(X � (0,T))]d.

By introducing a finite dimensional space Vd which is a suitableapproximation of V, the semi-discrete approximation of (6) reads :"t 2 (0,T] find ud = ud(t) 2 Vd such that

dttðqud;vÞX þAðud;vÞX ¼ LðvÞ 8v 2 Vd: ð7Þ

In the next section we will explain how to construct Vd for two dif-ferent families of non-conforming domain decomposition methods,namely, the Mortar Spectral Element Method (MSEM) and the Dis-continuous Galerkin Spectral Element Method (DGSEM). Bothmethods are well suited to allow: (1) variable approximation or-ders, that is an elementwise polynomial degree, (2) unstructuredand non-matching meshes, and (3) exponential rates of conver-gence in case of smooth solutions, [21].

3. Non-conforming Galerkin spectral formulations

In a domain decomposition approach we start by a discretiza-tion of the spatial differential operators in X, that relies on atime-independent three-level spatial decomposition of the domainX, as follows. At the first level, we subdivide X into K non


overlapping regions Xk, k = 1, . . . ,K, such that X ¼SK

k¼1Xk withXk \X‘ = ; if k – ‘ and we define the skeleton of this (macro)decomposition as S ¼

SKk¼1@Xk n @X. Note that this (macro) decom-

position can be geometrically non-conforming, i.e., for two neigh-bouring subdomains Xk, X‘, the interface c = @Xk \ @X‘ may notbe a complete side (for d = 2) or face (for d = 3) of Xk or X‘. Thenproblem (1) is solved in each Xk together with transmission condi-tions to ensure that the local solution is the restriction toXk � (0,T] of the global solution. For the elastic problem (1) thetransmission conditions read: (TC1) sut = 0 and (TC2) srt = 0,where s t denotes the jump of a quantity across a given interface.

To get the second level, in each Xk we introduce a partitioningT hk

, made by elements Xjk (quadrilaterals if d = 2 or hexahedras if

d = 3), with typical linear size hk and Xk ¼SJk

j¼1Xjk (see Fig. 1). Let

us set bX ¼ ð�1;1Þd and suppose that there exists a suitable invert-ible mapping Fj

k : bX ! Xjk with (positive) Jacobian JXj

k. This (meso)

partition is instead geometrically conforming in each Xk, thusthe intersection of two elements Xj

k; X‘k; ‘ – j, is either empty, or

a vertex, or an edge, or a face of both Xjk and X‘

k. We thus have thatZXk

f ¼X

Xjk2T hk

ZXj

k

f ¼X

Xjk2T hk

ZXðf � Fj

kÞJXjk:

The third (micro) level will be represented by the so-called Gass–batto–Legendre (GLL) points in each mesh element Xj

k. Let Q NkðbXÞ

be the space of functions defined on bX that are algebraic polynomi-als of degree less than or equal to Nk P 2 in each variable x1, . . . ,xd,and

Q NkðXj

kÞ ¼ v ¼ v � Fjk

�1: v 2 Q Nk

ðbXÞn o:

We define the finite dimensional space

XdðXkÞ ¼ vd 2 C0ðXkÞ : vdjXj

k2 Q Nk

ðXjkÞ; 8X

jk 2 T hk

n o;

and finally

Vd ¼ fvd 2 L2ðXÞh id

: vdjXk2 XdðXkÞ½ �d; 8k ¼ 1; . . . ;K : vdjCD ¼ 0g;

where d = {h,N} with h = (h1, . . . ,hK) and N = (N1, . . . ,NK) K-uplets ofdiscretization parameters. Each component hk and Nk representsthe mesh size and the degree of the polynomial interpolation inthe region Xk, respectively. In order to construct a nodal basis forVd, we introduce on each element Xj

k a set of interpolation points{pi} and corresponding degrees of freedom which allow to identifyuniquely a generic function in Vd. We remark the fact that, by the

Fig. 1. Example of a two dimensional subdomain partitioning. In this case K = 3 andX ¼ X1 [X2 [X3, with X1 ¼

S8j¼1X

j1; X2 ¼

S3j¼1X

j2 and X3 ¼

S2j¼1X

j3.

definition of the space Vd, the basis functions will not be globallycontinuous on the whole domain X. In the spectral element approx-imation, the interpolation points are the GLL points. On the refer-ence element bX, these points are tensor product of points definedin the interval [�1,1] as the zeros of ð1� x2ÞL0Nk

where L0Nkis the

derivative of the Legendre polynomial LNk. This means that there ex-

ist Nk + 1 points pi for the interpolation of a polynomial of degree Nk

in [�1,1] [14]. As previously observed, in higher dimensions, thespectral nodes {pi} are defined on the reference element bX via ten-sor product of the one dimensional distribution, and are thenmapped onto the generic element Xj

k in the physical space by Fjk.

In the SEM, the interpolation points are used also as quadraturepoints. Thus, we haveZbXðf � Fj

kÞJXjk

XðNkþ1Þd

i¼1

ðf � FjkÞðpiÞJXj

kðpiÞwi;

where wi are the weights of the GLL quadrature formula which isexact for all ðf � Fj

kÞJXjk2 Q 2Nk�1ðbXÞ. The spectral shape functions

Wi 2 Vd are defined as Wi(pj) = dij, i, j = 1, . . . , (Nk + 1)d, where dij isthe Kronecker symbol. It is straightforward to see that the restric-tion of any spectral function to Xj

k either coincides with a Lagrangepolynomial or vanishes. Moreover the support of any shape func-tion is limited to the neighbouring elements if the spectral node lieson the interface between two or more elements, while it is limitedto only one element for internal nodes.

To introduce the non-conforming Mortar and DiscontinuousGalerkin variational formulation, we write the equilibrium equa-tions for a generic Xk, integrate it by parts and sum overXj

k 2 T hk. What we obtain is an equivalent form of Eq. (6). For each

t 2 (0,T], we now seek for a K-uplet (ud,1, . . . ,ud,K) of functions, onefor each subdomain Xk. Problem (7) is then equivalent to:"t 2 (0,T] find (ud,1(t), . . . ,ud,K(t)) 2 Vd such thatXK

k¼1

dttðqud;k;vkÞXkþAðud;k;vkÞXk

þ Bðud;k;vkÞ@Xkn@X

� �¼XK

k¼1

LðvkÞXk; ð8Þ

for all (v1, . . . ,vK) 2 Vd, where

Aðu;vÞXk¼ ðrðuÞ; eðvÞÞXk

; and

Bðu;vÞ@Xkn@X ¼ ðrðuÞ � n;vÞ@Xkn@X: ð9Þ

Depending on the chosen non-conforming approach, the functionalspace Vd is completed by additional conditions on ud,k, k = 1, . . . ,K,on the skeleton of the macro decomposition which ensure thatud,k is the restriction to Xk of ud 2 [H1(X)]d. The bilinear formBð�; �Þ may either be zero or gather all the contributionsðrðud;kÞ � nk;vkÞ@Xkn@X; k ¼ 1; . . . ;K , depending on the chosen ap-proach. In fact, TC1 is imposed by introducing a weak continuitycondition on S compatible with the considered formulations whileTC2 is enforced strongly. In both situations this lead to a stronglyconsistent numerical method. This means that the exact solutionsatisfies the numerical scheme for each choice of h and N, [33].

Eq. (8) represents the starting point to introduce the Discontin-uous Galerkin variational formulation and the Mortar variationalformulation. With both formulations we will be able to treat moregeneral situations like (i) geometric non-conformity and (ii) poly-nomial degree non-conformity.

In (i) the partitions T k and T ‘, of different regions Xk and X‘ canhave mesh sizes hk and h‘ significantly different: in fact, the prac-tical importance of the proposed methods for elastodynamicsproblems lies on the possibility of using computational grids withdifferent local mesh sizes to take into account sharp variations inthe physical parameters of the media.


Furthermore, the vertices of elements Xjk and Xi

‘ lying on theskeleton S do not necessarily have to match, not even in the casehk = h‘ (Fig. 2).

In (ii) we use different polynomial approximation degrees ineach region to get higher precision without refining too muchthe grid. Moreover, as we show in Section 6, it is evident that highorder methods do not significantly suffer from numerical disper-sion. The combination of (i) and (ii) yields approximated solutionsthat are both numerically accurate and computationally cheap.

Obviously, interface conditions other than those we considerare possible as well: an intuitive alternative is offered by pointwisematching conditions which require different spectral solutions tomatch on a particular set of points lying on S. The DiscontinousGalerkin or Mortar approach is preferred to the pointwise match-ing since it brings optimal convergence rate, which is not the casefor methods based on pointwise conditions (see [34] for the ellipticcase), without affecting significantly the computational cost.

In the sequel, we describe in detail the non-conforming meth-ods. To ease the presentation, we suppose that each partition T hk

of Xk consists in only one element, this means that each regionis a spectral element. The more general case follows from similararguments.

3.1. Discontinuous Galerkin spectral formulation

Before going into the detail of the Discontinuous Galerkin spec-tral formulation let us introduce some notation that will be usefulin the sequel. Let us subdivide the skeleton S in elementary com-ponents as follows:

S ¼[Mj¼1

�cj; with ci \ cj ¼ ;; if i – j;

Fig. 3. Non-conforming domain decomposition (left) and skeleton structur

Fig. 2. Example of non-conforming decomposition.

where each element �cj ¼ @XkðjÞ \ @X‘ðjÞ� �

n @X, for some different po-sitive integers k and ‘: this decomposition is unique (see Fig. 3).Next we collect all the edges (faces if d = 3) in the set F I.

For any pair of neighbouring regions Xi and Xj that share a nontrivial edge (face) c 2 F I , we denote by vi, ri (resp. vj, rj) therestriction to Xi (resp. Xj) of regular enough functions v, r. We alsodenote by ni (resp. nj) the exterior unit normal to Xi (resp. Xj).

On each c 2 F I we define the average and jump operators for vand r as follows:

fvg ¼ 12ðvi þ vjÞ; svt ¼ vi ni þ vj nj; ð10Þ

and

frg ¼ 12ðri þ rjÞ; srt ¼ ri � ni þ rj � nj; ð11Þ

where a b 2 Rd�d is the tensor with entries(a b)ij = aibj,1 6 i,j 6 d, for all a;b 2 Rd.

After integration by parts over each region, the application ofjump and average operators defined in (10) and (11) and the impo-sition of condition TC2, i.e., continuity of tractions across S, we de-duce thatXK

k¼1

rðuÞ � n;vð Þ@Xkn@X ¼XM

j¼1

frðuÞg; svtð Þcj: ð12Þ

Since also TC1 holds, i.e., sut = 0 is zero across S, we can further addother terms in (12) that penalize and control the jumps of thenumerical solution, such as

hXM

j¼1

sudt; frðvÞgð ÞcjþXM

j¼1

gcjsudt; svtð Þcj

;

with h = {�1,0,1} and gcjpositive constants depending on the dis-

cretization parameters h and N and on the Lamé coefficients. Theterms do not affect consistency of the method and are added withthe purpose of providing more generality and better stability prop-erties to the scheme (see [11,35]).

In this context we choose gcj¼ afkþ 2lgA N 2

j =hj, where {q}A

represents the harmonic average of the quantity q, defined byfqgA¼2qkðjÞq‘ðjÞ=ðqkðjÞ þq‘ðjÞÞ; Nj¼maxðNkðjÞ;N‘ðjÞÞ; hj¼minðhkðjÞ;h‘ðjÞÞand a is a positive constant at disposal. The semi-discrete DGformulation reads:

"t 2 (0,T] find ud ¼ ðud;1ðtÞ; . . . ;ud;KðtÞÞ 2 VDGd � Vd such thatXK

k¼1

dtt qud;vð ÞXkþAðud;vÞXk

� �þXM

j¼1

Bðud;vÞcj¼XK

k¼1

LðvkÞ; ð13Þ

for all v ¼ ðv1; . . . ;vKÞ 2 VDGd , with

e (right) showing the elementary components (dark continuous lines).


Bðu;vÞcj¼ � frðuÞg; svtð Þcj

þ h sut; frðvÞgð Þcjþ gcj

sut; svtð Þcj:

ð14Þ

Corresponding to different values of h we obtain different DGschemes, namely: h = �1 (resp. h = 1) leads to the symmetric inte-rior penalty method SIPG (resp. non-symmetric NIPG), while h = 0corresponds to the so-called incomplete interior penalty methodIIPG (see [10,11,35,12] for more details).

3.2. Mortar spectral formulation

In this section we introduce the Mortar Spectral Element Meth-od for the solution of (8). The emphasis is on the numerical formu-lation, implementation and on the illustration of its flexibility andaccuracy. To illustrate the key points, we consider the free-vertexvariant of the MSEM [36,37]. The constrained-vertex strategy canbe implemented in a similar framework. For this latter techniquethe theoretical analysis is given in [8,34,38].

The MSEM relaxes the H1-continuity requirements of the con-forming spectral-element method by considering each element(or in the general case each macro region) individually and achiev-ing matching or patching conditions through a variational process.The mortars play the role of gluing the bricks of the spectral con-struction. Through the use of mortars, one can also couple domainswhere spectral elements are employed with others treated by fi-nite elements [34]. However, in this context we focus on non-con-forming spectral methods.

To begin, we denote by C‘k; ‘ ¼ 1; . . . ;2d, the edges (faces) of

each subdomain Xk, k = 1, . . . ,K, so that

@Xk ¼[2d

‘¼1

C‘k:

We then identify the skeleton S as the union of elementary non-empty components called mortars (or masters), more precisely

S ¼[Kk¼1

ð@Xk n @XÞ ¼[M

m¼1

�cm; with cm \ cn ¼ ;; if m – n;

where each mortar is a whole edge (or face) C‘ðmÞkðmÞ of a specific ele-

ment Xk(m) and m is an arbitrary numbering m = 1, . . . ,M, with Ma positive integer. Those edges or faces C‘

k that do not coincide witha mortar are called non-mortars (or slaves) and provide a dualdescription of the skeleton, as

S ¼[

m mortar

cþm ¼[

n non-mortar

c�n :

The intersection of the closures of the mortars defines a set of ver-tices or cross-points

Fig. 4. Non-conforming domain decomposition (left) and skeleton structure (right) shownon-mortars (dark dashed lines).

V ¼ xq ¼ �cþr \ �cþs� �

; xq R �cþm; m ¼ 1; . . . ;M� �

;

where q is an arbitrary numbering q = 1, . . . ,V. We define as well theset eV of virtual vertices (that are not cross-points) aseV ¼ ~xq ¼ ð�cþr \ cþs Þ

� �;

where q is an arbitrary numbering q ¼ 1; . . . ; eV (see Fig. 4).We define KdðC‘

kÞ ¼ Q NkðC‘

kÞ, the space of the traces of functionsof Xd(Xk) over C‘

k and we also introduce bKdðC‘kÞ ¼ Q Nk�2ðC‘

kÞ.We can now define the non-conforming spectral element dis-

cretization space eV d as the space of functions vd 2 Vd that satisfythe following additional mortar matching condition:

(MC1) let U be the mortar function associated with vd, i.e., afunction that is continuous on S, zero on @X and such that oneach mortar cm ¼ C‘ðmÞ

kðmÞ it coincides with the restriction ofvd;k ¼ vdjXk

to cm; then, for all indices (k,‘) such that C‘k is con-

tained in S but (k,‘) – (k(m),‘(m)) for all m = 1, . . . ,M (that isfor all indices (k,‘) such that C‘

k is a non-mortar) we require that:

ing a cro

ZC‘

k

ðvd;k �UÞ � bU ¼ 0 8 bU 2 bKdðC‘kÞ

h idð15Þ

and that

vdjXkðxqÞ ¼ UðxqÞ; 8xq 2 V [ eV : ð16Þ

The integral matching condition (15) represents a minimization ofthe jump of the functions at internal boundaries with respect to theL2-norm and is the counterpart in the Mortar framework of condi-tion TC1. The vertex condition (16) ensures exact continuity atcross-points. The Mortar spectral formulation is obtained by solv-ing in each region Xk the elastodynamics variational problem (8)with homogeneous Neumann boundary conditions on S(r(u) � n = 0 so that

PkB u;vð Þ@Xkn@X is identically zero, i.e., TC2 is

satisfied), and enforcing weak continuity of the displacement onS with mortar condition (15).

Thus, the semi-discrete Mortar spectral formulation reads:"t 2 (0,T] find ðud;1ðtÞ; . . . ;ud;KðtÞÞ 2 Vmortar

d such that

XK

k¼1

dtt qud;k;vk

� �XkþAðud;k;vkÞXk

¼XK

k¼1

LðvkÞ; ð17Þ

for all ðv1; . . . ;vKÞ 2 Vmortard where

Vmortard ¼ ðv1; . . . ;vKÞ 2 Vd : mortar condition MC1 is satisfiedf g:

The Mortar Element Method was originally proposed as a non-over-lapping domain decomposition approach, however recently it hasbeen generalized to the case of overlapping subdomains [39–41].

ss-point (�), a virtual vertex (h), the mortars (dark continuous lines) and the

Ω2

Ω1

Ω2

Ω1

S

S

A

B

C

D

Fig. 5. Example of rectangular domain X where the surface S separates two different physical materials. Non overlapping subdomains and meshes (left), holes as non meshedregions (shadowed areas) and overlapping subdomains and meshes (right).


The overlapping version may be quite useful in elastodynamicsmodelling to treat subdomains with complex shaped boundaries(see an application in Section 8). Let us consider the two cases pre-sented in Fig. 5.

On the one hand, the rectangular domain X is partitioned intotwo non-overlapping subdomains X1, X2 and the skeleton S ofthe decomposition coincides with the separation surface betweentwo different materials such that elastic waves propagate fasterin X2 than in X1. By adapting the mesh size hk in each subdomainXk according to the propagation velocity of the elastic waves in thesubdomain, one reasonably selects h1 > h2. However, h2 has to besmall enough to follow the shape of S and h1 cannot be too largeotherwise some holes appear close to the surface S. As a conse-quence, h1 h2 in a neighbourhood of S resulting in a large num-ber of unknowns to consider in both subdomains. The mortarmatching condition allows to transfer the displacement from theset of master interpolation points to the set of slave ones and bothsets of points are located on the (d-1)-dimensional surface S. Notethat numerical results are independent of the choice of the masterand of the slave subdomains.

On the other hand, the rectangular domain X is partitioned intotwo overlapping subdomains, namely, X1 which is the bottom left-handed region under the dashed polyhedrical surface AB and X2

the top right-handed region over the solid line S. These two subdo-mains overlap in the region between S and the surface AB. In thiscase, we can have h1 > h2 everywhere in X1 and the mortar match-ing condition allows to transfer the displacement from the set ofinterpolation points of X1 which are contained in the d-dimen-sional region bounded by the polyhedrical surfaces AB and CD tothe set of interpolation points of X2 which are on the (d-1)-dimen-sional surface S. Indeed, in the overlapping case, the slave subdo-main always covers the master one. Moreover, the slavesubdomain is chosen as the one where the mesh best describesthe surface S and the master subdomain contains the source ofelastic waves. In the overlapping case, the matching conditionreads:

(MCO1) let U be a function that is equal to vd,k in the d-dimen-sional elements of master subdomain Xk containing a part of S,and zero elsewhere. Then, for each slave subdomain Xi suchthat @Xi \ S – ;, we require that:
Z
Sðvd;i �UÞ � bU ¼ 0 8 bU 2 ½Kd;iðSÞ�d; ð18Þ

where Kd;iðSÞ is the space of the traces over S of functions belongingto Xd(Xi).

4. Algebraic formulations and time integration

We discuss here the algebraic formulations of the two non-con-forming approaches presented in the previous sections. In particu-lar we describe how to construct the linear system resulting from

the Mortar or the DG discretizations and subsequently we intro-duce the time integration scheme employed for the numericalsimulations.

4.1. Algebraic formulation of the problem

We consider the elastodynamics equation (1) in a bounded re-gion X � R2 with Dirichlet boundary condition, thus CD � @X. Toease the presentation let also suppose that X is partitioned intoK non-overlapping spectral elements X1, . . . ,XK so that S ¼TK

k¼1@Xk n CD. The more general case can be obtained in a similarmanner.

We denote by D ¼PK

k¼1ðNk þ 1Þ2 the dimension of each compo-nent of Vd and we introduce a basis fW1

i ; W2i g

Di¼1 for the finite

dimensional space Vd, where W1i ¼ ðW

1i ;0Þ

> and W2i ¼ ð0;W

2i Þ>.

Dropping the subscript d, we write the trial functions u 2 Vd aslinear combination of basis functions

uðx; tÞ ¼XD

j¼1

W1j ðxÞ0

" #U1

j ðtÞ þXD

j¼1

0W2

j ðxÞ

" #U2

j ðtÞ; ð19Þ

Next, we define ak ¼ 1þPk�1

j¼1 ðNj þ 1Þ2 and bk ¼Pk

j¼1ðNj þ 1Þ2 andwe order the basis functions such that

ujXk¼ ðu1;u2Þ>jXk

¼Xbk

j¼ak

W1j U1

j;k;Xbk

j¼ak

W2j U2

j;k

!>; ð20Þ

for k = 1, . . . ,K. With the notation just introduced, we write the Eq.(8) for any test function W‘

j ðxÞ, for ‘ = 1, 2, in the space Vd and weobtain the following set of discrete ordinary differential equations:

M€Uþ AUþ BU ¼ Fext; ð21Þ

or equivalently

M1 00 M2

" #€U1

€U2

" #þ A1 þ B1 A2 þ B2

A3 þ B3 A4 þ B4

" #U1

U2

" #¼ Fext;1

Fext;2

" #; ð22Þ

where €U represents the vector of nodal accelerations and Fext thevector of externally applied loads. As a consequence of assumptionson the basis functions, the mass matrices M1 and M2 have a blockdiagonal structure M‘ ¼ diagðM‘

1;M‘2; . . . ;M‘

KÞ, for ‘ = 1, 2, whereeach block M‘

k is associated to the spectral element Xk and

M‘kði; jÞ ¼ ðqW‘

j ;W‘i ÞXk

; for i; j ¼ ak; . . . ; bk: ð23Þ

The matrix A associated to the bilinear form Að�; �Þ defined in (9)takes the form

A ¼ A1 A2

A3 A4

" #;

where the block diagonal matrices A‘, for ‘ = 1, . . . ,4 are equal toA‘ ¼ diag A‘

1;A‘2; . . . ;A‘

K

� �. The elements of the matrices A‘

k, for‘ = 1, . . . ,4 and k = 1, . . . ,K are defined by


A1kði; jÞ ¼ A rðW1

j Þ; eðW1i Þ

� �Xk

; A2kði; jÞ ¼ A rðW2

j Þ; eðW1i Þ

� �Xk

;

A3kði; jÞ ¼ A rðW1

j Þ; eðW2i Þ

� �Xk

; A4kði; jÞ ¼ AðrðW

2j Þ; eðW

2i ÞÞXk

;

ð24Þ

for i, j = ak, . . . ,bk. We remark that the matrices M and A are verysimilar to those resulting from the discretization of the elastody-namics equation (6) with conforming methods like the Spectral Ele-ment Method (see [13,14]).

The matrix B, associated to the bilinear form Bð�; �Þ defined in(9), is the one that takes into account the discontinuity of thenumerical solution across the skeleton S. In the DG approach it isexpressed by

B ¼ B1 B2

B3 B4

" #;

where

B‘ ¼

B‘1;1 � � � B‘

1;K

..

. . .. ..

.

B‘K;1 � � � B‘

K;K

26643775; for ‘ ¼ 1; . . . ;4:

In particular the elements of each matrix B1k;n are defined by:

B1k;nði; jÞ ¼

Xc2F I

BðW1j ;W

1i Þc ¼

Xc2F I

�Z

crðW1

j Þn o

: sW1i t

þ hZ

csW1

j t : rðW1i Þ

� �þ gc

Zc

sW1j t : sW1

i t;

for i = ak, . . . ,bk and j = an, . . . ,bn. The elements of the matrices B‘k;n,

for ‘ = 2, 3, 4 are defined in a similar way.The situation is a little bit more complicated in the Mortar ap-

proach, since the weak continuity condition across the skeleton Sdoes not appear explicitly in the variational equation but it is aconstraint in the functional space Vmortar

d : in fact, in the Mortar Var-iational Formulation, Bð�; �Þ ¼ 0 implies that B is a null matrix.

To account for MC1 we need to modify (22) as follows. Withoutloss of generality let us suppose that c�n is a non-mortar edge con-tained in S and moreover that it is shared by two regions Xm andXn. We call master the side of c�n belonging to Xm and slave theother side. Thus, the mortar conditions MC1 can be recast as:

(i) U = um on c�n ,(ii)

Rc�nðun � umÞ � bU ¼ 0 8 bU 2 bKdðc�n Þ

h id.

We remark that when interfaces do not match geometrically,i.e. c�n is shared by M⁄ + 1 regions Xn;Xmð1Þ; . . . ;XmðM�Þ, condition(ii) reads as

XM�‘¼1

Zc�n \@Xmð‘Þ

ðun � umð‘ÞÞ � bU ¼ 0 8 bU 2 bKdðc�n Þh id

:

Then the following arguments have to be intended for each integralin the above expression. Now, for the spectral element Xn (resp.Xm) we order first the Nn + 1 (resp. Nm + 1) degrees of freedom(d.o.f.) associated to the spectral nodes pi that live in c�n and nextthe d.o.f. associated to the remaining spectral nodes pi. With theseassumptions the restriction of the function un on c�n is rewritten as

un jc�n¼

XNnþ1

j¼1

W1j U1

j;n;XNnþ1

j¼1

W2j U2

j;n

!T

;

and the same for the function um jc�n. Hence, by definition of scalar

product, the mortar condition (ii) becomes

Zc�n

ðu1n � u1

mÞbU1 þZ

c�n

ðu2n � u2

mÞbU2 ¼ 0 8bU1; bU2 2 bKdðc�n Þ: ð25Þ

Since the integrals in (25) concern separately the two componentsof the displacement, we focus the attention onto one of them, drop-ping the superscripts to ease the notation. The other one is treatedin the same manner. For the slave side of the mortar we obtainZ

c�n

unbUi ¼

XNnþ1

j¼1

Uj;n

Zc�n

WjbUi ¼

XNnþ1

j¼1

Ri;jUj;n; for i ¼ 1; . . . ;Nn � 1;

ð26Þ

where Ri;j ¼Rc�n

WjbUi. For the master side, using the mortar condition

(i), we have thatZc�n

umbUi ¼

XNmþ1

j¼1

Uj;m

Zc�n

UjbUi ¼

XNmþ1

j¼1

Pi;jUj;m; for i ¼ 1; . . . ;Nn � 1;

ð27Þ

with Pi;j ¼Rc�n

UjbUi. One may use (26) and (27) to recast the mortar

constraint MC1 in matrix notation

R

U1;n

..

.

UNnþ1;n

26643775 ¼ P

U1;m

..

.

UNmþ1;m

26643775

Now, to compute numerically the matrices R and P we use suitablequadrature formulas depending if we are on the slave or in the mas-ter side of the mortar. We choose Nn + 1 GLL nodes to evaluate theintegrals

Rc�n

WjbUids such that the matrix R takes a special structure.

In fact, because of this choice the interior part Rint is diagonal. Thefirst and the last columns are full but they are concerned only withd.o.f. (namely, U1,n and UNnþ1;n) but do not depend on the matchingconditions. We observe also that the matrix P is full. Then the localprojection operator can be written in a matrix form as

U2;n

..

.

UNn ;n

26643775 ¼

R�1int

P1;1 � � � P1;Nmþ1 �R1;1 �R1;Nnþ1

..

. . .. ..

. ... ..

.

PNn�1;1 � � � PNn�1;Nmþ1 �RNn�1;1 �RNn�1;Nnþ1

26643775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Qn

U1;m

..

.

UNmþ1;m

U1;n

UNnþ1;n

266666664

377777775:

Thanks to the projection operator Qn, we are then able to recoverthe slave unknowns in c�n once we know the master ones. To obtaina global projection operator eQ we proceed as follows. For each com-ponent of u we denote by Uslave the vector of unknowns associatedto d.o.f. that lay on the slave side of S and by Umaster the vector ofunknowns associated to all the remaining d.o.f. Then, for each c�nbelonging to the skeleton S we build the local projection operatorQn and we store it into the matrix eQ . In this way eQ has a blockstructure of the form

eQ ¼ bQ 0

0 bQ24 35; ð28Þ

where bQ is a block diagonal matrix with a block equal to the iden-tity and the other equal to the rectangular matrix Q containing allthe local matrices Qn. Thus, we have that the global linear systemcan be expressed aseQ >fM eQ €Umaster þ eQ >eA eQ Umaster ¼ eQ >Fext; ð29Þ


where the matrices fM and eA have columns and rows modified withrespect to the ones of M and A according to latter assumptions onthe unknowns reordering. All the terms appearing in the matricesof the two algebraic formulation are computed using Gauss–Lobattoquadrature rule in which the quadrature points coincide with theGLL points. We remark that since the term W‘

j W‘i 2 QNk

, for somek, while the Gauss–Lobatto rule with Nk points is exact for polyno-mials up to degree 2Nk � 1 in each variable, the spectral mass ma-trix M is slightly under integrated. However, the final accuracy ofspectral methods is maintained [13].

4.2. Structural damping

When using Eq. (5) to model viscoelastic materials, very usefulfor seismic applications, we must compute additional externalforces:

Fvisc ¼ �C _U� DU;

or equivalently

Fvisc;1

Fvisc;2

" #¼ � C1 0

0 C2

" #_U1

_U2

" #� D1 0

0 D2

" #U1

U2

" #;

where the matrices C‘ and D‘, for ‘ = 1, 2 are block diagonal. Eachblock C‘

k and D‘k is associated to the spectral element Xk and

C‘kði; jÞ ¼ ðqfW‘

j ;W‘i ÞXk

; D‘kði; jÞ ¼ ðqf2W‘

j ;W‘i ÞXk

; ð30Þ

respectively for i, j = ak, . . . ,bk. Then the final discretized systembecomes:

M€Uþ C _Uþ ðAþ Bþ DÞU ¼ Fext; ð31Þ

where the accelerations €U and the velocities _U are approximated asdescribed in the following section.

4.3. Time integration scheme

Let now subdivide the interval (0,T] into N subinterval of ampli-tude Dt = T/N: at every time level tn = nDt, for n = 0, . . . ,N, the timeintegration scheme is achieved with the second order Leap-Frogscheme (cf. [31]):

MUðtnþ1Þ ¼ 2M� Dt2ðAþ BÞ �

UðtnÞ �MUðtn�1Þ þ Dt2FextðtnÞ;ð32Þ

oreQ >fM eQ Umasterðtnþ1Þ

¼ eQ > 2fM�Dt2eA� � eQ UmasterðtnÞ �fM eQ Umasterðtn�1Þ þDt2FextðtnÞh i

;

ð33Þ

respectively for (22) and (29), with initial conditions U(t0) = u0 and_Uðt0Þ ¼ u1. In particular if a DGSEM is employed the iteration ma-trix M in (32) is diagonal and can be inverted at very low computa-tional cost. In the MSEM the matrix eQ >fM eQ is non-diagonal, buttaking advantage of the structure of eQ it is possible to split the lin-ear system (33) as follows

Mmaster 00 Q>MslaveQ

" #UImasterðtnþ1ÞUSmasterðtnþ1Þ

" #¼

bImaster

Q>bSslave

" #; ð34Þ

with b ¼ 2fM � Dt2eA� � eQ UðtnÞ �fM eQ Uðtn�1Þ þ Dt2FextðtnÞh i

. Herethe superscripts I and S identify those unknowns belonging respec-tively to the interior or to the skeleton of the domain. Then at eachtime step we solve separately the two blocks of the linear system(34). In particular for the non-diagonal block we perform the LU-factorization (see [33]).

To ensure stability, the explicit time integration scheme mustsatisfy the usual Courant–Friedrichs–Levy (CFL) condition (see[28]) that imposes a restriction on Dt. We see in Section 6 that thislimitation is proportional to the minimal distance between twoneighbouring spectral nodes of the numerical grid. Since this dis-tance scales as hkN�2

k (hk size of the spectral element Xk), the sta-bility requirement on Dt may become too restrictive for verylarge polynomial degrees Nk. For these cases an implicit timescheme is recommended.

5. Error estimates

In this section we introduce some notation and present a priorierror estimates for the semi-discrete formulations (13) and (17)respectively. For the DG formulation (13), we show, in a suitablemesh-dependent energy norm, error estimates that are optimalwith respect to the mesh size h and suboptimal with respect tothe polynomial approximation degree N. Such results are in agree-ment with those proved in [12,35,15] for a slightly different DGmethod.

For the MSEM formulation (17), in agreement with [8,19], weprove an optimal error bound in both h and N, using the H1-brokennorm.

At the end of this section we also state a priori error estimatesfor the fully discrete problems (32) and (34) respectively, obtainedusing the above results and standard techniques. To ease the pre-sentation, we postpone the proofs of the convergence results andall the technical details to Section 9.

For the error analysis we consider the problem (6) defined inX � R2 with @X = CN [CD. We suppose that its unique solutionu is regular enough so that all the norms we introduce are well de-fined. Moreover, in the following, C will denote a positive constantthat varies at each occurrence but is independent of the discretiza-tion parameters h and N. We also assume X to be partitioned intoK non overlapping quadrilaterals X1, . . . ,XK and that S is subdi-vided in M elementary components c1, . . . ,cM (resp. non-mortaredges c�1 ; . . . ; c�M) for the DGSEM case (resp. for the MSEM case).The more general situation can be obtained using similararguments.

For Xk � X we denote by k � kp;Xk(resp. j � jp;Xk

) theHp(Xk) = [Hp(Xk)]d norm (resp. seminorm). When Xk = X we sim-ply write k � kp = k � kp,X (resp. j � jp = j � jp,X). Since we are dealingwith time dependent functions, we take the standard approach oftreating these as maps from a time interval into a Banach space Xand set

kukLpð0;t;XÞ ¼Z t

0kukp

X

� 1=p

; 0 6 t 6 T; 1 6 p 61;

with the obvious modifications when p =1.

5.1. Semi-discrete error estimates - DGSEM

To analyze the DG formulation (13) we introduce the enrichedspace VðdÞ ¼ Vd � ðH2ðXÞ \H1

CDðXÞÞ and define the following

mesh-dependent norms:

kvdkDG ¼XK

k¼1

kD1=2 eðvdÞk20;XkþXM

j¼1

gcjksvdtk2

0;cj

!1=2

8vd

2 Vd; ð35Þ

and

jjjvjjjDG ¼ kvk2DG þ

XK

k¼1

hk

N2k

!2

jvj22;Xk

0@ 1A1=2

8 v 2 VðdÞ: ð36Þ


Notice that, when restricted to Vd, these two norms are uniformlyequivalent, thanks to a local inverse inequality [10]. We also set

ADGðu;vÞ ¼XK

k¼1

Aðu;vÞXkþXM

j¼1

Bðu;vÞcj8 u;v 2 VðdÞ; ð37Þ

A�DGðu;vÞ ¼ ADGðu;vÞ þXM

j¼1

gcjs@tut; svtð Þcj

8 u;v 2 VðdÞ: ð38Þ

Notice that formulation (38) was introduced in [12,15]. We willmake use of the auxiliary form (38) to prove optimal error estimatein h and suboptimal in N for the DG scheme. In the sequel we willuse the results in [12,15] to complete the analysis. All the details aregiven in Section 9.

Lemma 1. There exist two positive constants M and j such that:

ADGðu;vÞ 6 Mjjjujjj2DGjjjvjjj2DG 8u;v 2 VðdÞ; ð39Þ

ADGðud;udÞP jkudk2DG 8ud 2 Vd: ð40Þ

For h = 0, � 1 the last inequality holds provided that gcjis chosen suf-

ficiently large 8cj 2 F I .

Proof. Inequality (39) follows from the definition of the jjj � jjjDG-norm (35) by applying the Cauchy–Schwarz and trace inequality.If h = �1 (40) holds with j = 1 (and is indeed an equality). Ifh = �1 or 0 we observe that by the inverse-trace inequality weget 8cj 2 F I : cj � @Xk

frðuÞgk k20;cj6 C

N2j

hjkuk2

DG þh2

k

N4k

juj22;Xk

!6 C

N2j

hjkuk2

DG 8u 2 Vd;

For all 0 6 t 6 Tsee [10]. Setting d� ¼minjfkþ 2lgA;cjwe deduceXM

j¼1

frðuÞg; sutð Þcj

�� 6 Cad�kuk2

DG

XM

j¼1

gcjksutk2

0;cj6

Cad�kuk2

DG:

Then, it holds

ADGðu;uÞP kuk2DG � 2

XM

j¼1

frðuÞg; sutð Þcj

�� P ð1� 2C=ad�Þkuk2

DG:

Choosing a sufficiently large such that 1 � 2C/ad⁄ is bounded awayfrom zero we have (40). h

Now, for all 0 6 t 6 T we set uDG = uDG(t) the unique solution inVd of the problem,

dtt quDG;vð Þ þ ADGðuDG;vÞ ¼ LðvÞ 8v 2 Vd: ð41Þ

From Lemma 1 and standard techniques, follows that the varia-tional problem in (41) is well posed. For uDG we have the followingconvergence result. For the proof see Section 9.

Theorem 1. There exists a positive constant C such that

supt2½0;T�

jjjðu� uDGÞðtÞjjjDG 6 CXK

k¼1

h2mk�2k

N2sk�3k

kuk2H2ð0;T;Hsk ðXkÞÞ

( )1=2

; ð42Þ

where Nk P 1 and mk = min(Nk + 1,sk).

5.2. Semi-discrete error estimates - MSEM

We now move on the error analysis for the MSEM semi-discret-ization (17). Let H1/2(@Xk) be the trace space of H1(Xk) on @Xk, en-dowed with the norm

kuk1=2;@Xk¼ kuk2

0;Xkþ juj21=2;@Xk

� �1=2;

with juj1=2;@Xk¼ minvj@Xk

¼ujvj1;Xk, and for any c � @Xk, define the

space H1=200 ðcÞ as

H1=200 ðcÞ ¼ u 2 H1=2ðcÞ : u 2 H1=2ð@XkÞ

n o;

where u is the extension by zero of u to @Xk, see [30]. Moreover weintroduce the mesh-dependent norm

kuk� ¼XK

k¼1

kuk21;Xk

!1=2

8u 2 VðdÞ;

and we define the bilinear form AMð�; �Þ by

AMðu;vÞ ¼XK

k¼1

Aðu;vÞXk8u;v 2 VðdÞ:

We have the following properties for AMð�; �Þ.

Lemma 2. There exists two positive constants M and j, independentof h and N, such that

AMðu;vÞ 6 Mkuk�kvk� 8u;v 2 VðdÞ; ð43ÞAMðud;udÞP jkudk2

� 8ud 2 Vmortard : ð44Þ

Proof. Inequality (43) is a direct consequence of the Cauchy–Sch-warz inequality, while (44) is easily obtained using the generalizedKorn’s first inequality and the Poincaré inequality for ud 2 Vmortar

d ,see [42]. h

For all 0 6 t 6 T let uM = uM(t) be the solution of the variationalproblem (17) in Vmortar

d or equivalently of

q@ttuM;vð Þ þ AMðuM;vÞ ¼ LðvÞ 8v 2 Vmortard : ð45Þ

For uM it holds the following convergence result. For the proof seeSection 9.

Theorem 2. There exists a positive constant C such that

supt2½0;T�

kðu� uMÞðtÞk� 6 CXK

k¼1

h2mk�2k

N2sk�2k


( )1=2

; ð46Þ


5.3. Fully-discrete error estimates – DGSEM/MSEM

As mentioned at the beginning of this section, we sketch theproof of the fully discrete error estimates form the DGSEM andMSEM formulations. We recall that the discrete formulation ofproblems (13) and (17) is obtained by approximating the secondorder time derivative with the Leap-Frog scheme as in Section 4.3.Then, at each time step tn = nDt, n P 2, the problem reads: findud(tn) 2 Vd such thatXK

k¼1

1Dt2 q udðtnÞ � 2udðtn�1Þ þ udðtn�2Þð Þ;vð ÞXk

þXK

k¼1

Aðudðtn�1Þ;vÞXk

þXM

j¼1

Bðudðtn�1Þ;vÞcj¼XK

k¼1

LðvkÞ 8v 2 Vd ð47Þ

for the DGSEM case, and: find udðtnÞ 2 Vmortard such thatXK

k¼1

1Dt2 q udðtnÞ � 2udðtn�1Þ þ udðtn�2Þð Þ;vð ÞXk

þXK

k¼1

Aðudðtn�1Þ;vÞXk

¼XK

k¼1

LðvkÞ 8v 2 Vmortard ð48Þ


for the MSEM case, respectively. Comparing the fully discrete solu-tion (47), resp. (48), with the semi-discrete solution (13), resp. (17),using estimate (42), resp. (46), and standard techniques it is possi-ble to prove the following result.

Theorem 3. Suppose that u0, u1, f, t and the solution u of (1) aresufficiently smooth. Then, there exists a constant C = C(u0,u1,f,t,u)such that "n P 2 it holds

q1=2ðutðtnÞ � @tuDGðtnÞÞ��

0 þ uðtnÞ � uDGðtnÞk kDG

6 C Dt2 þXK

k¼1

hmk�1k

Nsk�3=2k

!;

for the DGSEM, and

q1=2ðutðtnÞ � @tuMðtnÞÞ��

0 þ uðtnÞ � uMðtnÞk k�

6 C Dt2 þXK

k¼1

hmk�1k

Nsk�1k

!;

for the MSEM, where Nk P 1 and mk = min(Nk + 1,sk).

6. Analysis of grid dispersion and stability

In this section we study in detail the MSEM and the DGSEM inthe two dimensional case, doing the so called Von Neumann anal-ysis, namely the analysis of grid dispersion and stability. The for-mer criterion determines the lowest sampling ratio for thespatial discretization (i.e., the number of nodes per wavelength)such that the numerical solution has a prescribed accuracy. The lat-ter determines the largest time step Dt that we are allowed to usein the explicit time integration scheme, such that the solution re-mains bounded with respect to problem’s data. For the sake of sim-plicity, we present the dispersion and stability analysis in a twodimensional framework.

To start with, let us consider the wave equation (1) in an isotro-pic, elastic, unbounded domain X, with u(x, t) ? 0 for all t asjxj?1. We also assume f � 0: this is not a limitation. These arestandard assumptions when using the Von Neumann’s method(plane wave analysis), see [43,23,44,16,21,45–47].

At the discrete level we assume that X is partitioned into non-overlapping spectral elements Xk having uniform size h. This par-titioning is supposed to be periodic and made by squared elements

Fig. 6. Periodic grid made by squared elements with side parallel to the coordinateaxis. The reference element XC with sides cf and neighbouring elements Xf, forf = {R,L,T,B}.

with sides parallel to the coordinate axes (cf. Fig. 6). We also sup-pose the polynomial approximation degree equals N in each Xk.

6.1. Grid dispersion – DGSEM

We report the analysis of grid dispersion for the DGSEM, seealso [16] for the scalar case. Let identify by W‘;Xf ; ‘ ¼ 1;2 the basisfunctions with support in Xf with f 2 {C,T,B,L,R} (cf. Fig. 6). With-out loss of generality, we consider respectively test and trial func-tions of the following form

W‘i ¼

W‘;XCi in XC ;

0 otherwise;

(

and

W‘j ¼

W‘;XCj in XC ;

W‘;Xf

j in Xf ;

0 otherwise:

8>><>>: f 2 fT;B; L;Rg: ð49Þ

By rewriting Eq. (21), we obtain a rectangular linear system in theunknowns

U‘ ¼ U‘;XC ;U‘;XT ;U‘;XB ;U‘;XL ;U‘;XR

h i; ‘ ¼ 1;2: ð50Þ

Clearly this system is underdetermined because the number of col-umns, 10(N + 1)2, exceeds the number of rows, 2(N + 1)2. To reduceit into a square linear system we make use of the following planewave hypothesis.

Let us assume that the displacement is a plane wave, i.e., in XC

we have

U‘;XCj ¼ c‘j e

iðj�pj�xtÞ; ‘ ¼ 1;2; ð51Þ

where j = (kx,ky) is the wave vector, pj contains the jth node inCartesian coordinates and cj are arbitrary constants. The aboveassumption implies that

U‘;Xf

j ¼ ebf U‘;XCj ; ‘ ¼ 1;2; ð52Þ

with bf = {� ikyh, ikyh,� ikxh, ikxh} and f = {T,B,R,L}, respectively.Substituting (52) in (50) we obtain the modified square linear sys-tem of size 2(N + 1)2:

M1 00 M2

" #€U1;XC

€U2;XC

" #þ A1 þ eB1

A2 þ eB2

A3 þ eB3A4 þ eB4

24 35 U1;XC

U2;XC

" #¼

00

� �;

ð53Þwhere Mi, i = 1, 2, and Ai, i = 1, . . . ,4, are defined in (23) and (24)respectively. The matrices eB‘

, for ‘ = 1, . . . ,4, are defined taking intoaccount the hypothesis of periodicity of the discretization and theplane wave assumption (51); for example eB1

is given byeB1ði; jÞ ¼ B1ði; jÞ þ

Xf¼fT;B;R;Lg

ebf B1;f ði; jÞ; i; j ¼ 1; . . . ; ðN þ 1Þ2;

where

B1ði; jÞ ¼X

f¼fT;B;R;Lg�Z

cf

frðW1;XCj Þg : sW1;XC

i t

þ hZ

cf

sW1;XCj t : frðW1;XC

i Þg þ gf

Zcf

sW1;XCj t : sW1;XC

i t;

and

B1;f ði; jÞ ¼ �Z

cf

rðW1;Xf

j Þn o

: sW1;XCi tþ h

Zcf

sW1;Xf

j t : rðW1;XCi Þ

n oþ gf

Zcf

sW1;Xf

j t : sW1;XCi t:

Similarly, we define the elements of the matrices eB‘ for ‘ = 2, 3, 4.Now, calculating the second derivative with respect to time of


U‘;XC and setting eK ¼ Aþ eB we obtain the following generalizedeigenvalue problemeKUXC ¼ KMUXC ; ð54Þ

where K ¼ x2h , with xh the angular frequency at which the wave

travels in the grid. As observed in [17,16,48] the number of eigen-values of problem (54) naturally exceeds the number of admissiblephysical modes. Then we need a strategy to select which eigen-values correspond to the compressional (cP) and the shear (cS) wavevelocities. We do this by computing all the velocities associated tothe eigenvalues of (54) and then comparing them to the real cP

and cS velocities defined in (4).We denote by KP and KS the eigenvalues used to compute the

best approximations of cP and cS, namely cP,h and cS,h.Note that the system (54) for the NIPG method is not symmetric

thus complex eigenvalues are possible. However in [16] it has beenremarked that KP and KS are in fact always real numbers. Next wedefine the grid dispersion of pressure and shear waves as the ratiobetween the velocity at which the wave travels in the grid (numer-ical velocity) and the physical velocity. By definition the numericalshear velocity cS,h is given by cS,h = hxh/(2pd), where d = h/(NL) isthe sampling ratio (or equivalently d�1 is the number of GLL pointsper wavelength), and L is the wavelength of the plane wave. Wehave that cS;h ¼ h

ffiffiffiffiffiffiKSp

=ð2pdÞ, and therefore the grid dispersion isthe relative error in the velocity, given by eS = cS,h/cS � 1. Analo-gously cP;h ¼ h

ffiffiffiffiffiffiKPp

=ð2pdÞ and eP = cP,h/cP � 1.

6.2. Grid dispersion – MSEM

In order to carry out a dispersion analysis for the MSEM, weadopt a strategy similar to the one described for the DGSEM. Thegoal is the definition of a generalized eigenvalue problem associ-ated only to the degrees of freedom belonging to XC. Under thehypothesis of shape regularity and periodicity of the mesh, we ob-serve that the skeleton of the partitioning is uniquely defined oncethe master and slave edges for the reference element XC are se-lected. Consider the configuration shown in Fig. 7: this is the un-ique, up to a rotation, possible combination of master and slaveedges for XC that does not violate the hypothesis of gridperiodicity.

We rewrite the ODE system (21) in the MSEM framework: using(49) we obtain

Fig. 7. Periodic grid made by squared elements with side parallel to the coordinateaxis. The reference element XC with sides cf and neighbouring elements Xf, forf = {R,L,T,B}. Solid lines (–) are the master edges and dashed lines (--) are the slaveedges.

M€UXC þ AUXC ¼ 0; ð55Þ

where M and A are defined in (23) and (24), respectively. Next, weimpose the mortar conditions MC1 concerning the slave unknownsat the interfaces cB and cR. In particular, for ‘ = 1,2 and f = {R,B},we have thatXj:pj2cf

U‘;XCj

Zcf

W‘;XCjbU‘

i ¼X

j:pj2cf

U‘;Xf

j

Zcf

W‘;Xf

jbU‘

i 8bU‘i 2 bKdðcf Þ: ð56Þ

To recast these conditions in terms of unknowns and basis functionsdefined only on XC we simply notice that, by periodicity,

W‘;XRj jcR

¼ W‘;XCj jcL

and W‘;XBj jcB

¼ W‘;XCj jcT

; ‘ ¼ 1;2 ð57Þ

and by the plane wave assumption, Eq. (52) holds. Substituting (57)in (56) we obtainXj:pj2cf

U‘;XCj

Zcf

W‘;XCjbU‘

i ¼X

j:pj2cf

ebf U‘;XCj

Zcf

W‘;Xf

jbU‘

i 8bU‘i 2 bKdðcf Þ;

ð58Þ

for ‘ = 1, 2 and f = {B,R}. We remark that Eq. (58) relates slave un-knowns in XC with master unknowns still in XC: this means thatthe matrix projection eQ refers only to the reference element XC.We use this matrix to reduce the linear system in (55) to one forthe master unknowns onlyeQ >fM eQ €UXC

master þ eQ >eA eQ UXCmaster ¼ 0: ð59Þ

We notice that eQ has always a block diagonal structure like (28)where each block bQ is modified according to (58). Calculating thesecond derivative of the displacement with respect to time anddefining eK ¼ eQ > eA eQ , we finally obtain the generalized eigenvalueproblem of size 2(N2 + 3)eKUXC

master ¼ K eQ >fM eQ UXCmaster ; ð60Þ

where K ¼ x2h as in the DGSEM case.

We remark that in the definition of eS and eP the sign of the errorindicates if the numerical approximation causes a delay or anacceleration of the travelling waves. The grid dispersion error willdepend on the sampling ratio d, the wave vector j, the degree ofthe basis function N and on the velocities cP and cS. For the DGSEM,on the stability parameter g too.

6.3. Grid dispersion – numerical results

Now, we analyze the grid dispersion error for both the MSEMand the DGSEM from three different points of view: (i) the conver-gence with respect to the polynomial degree N, (ii) the conver-gence with respect to the sampling ratio d and (iii) the numericalanisotropy introduced by the grid dispersion. Finally we comparethe results with the conforming SEM: in this case the grid disper-sion analysis is obtained using a technique similar to the one em-ployed in the MSEM and the results obtained are in agreementwith [16,22].

In the first set of experiments, we fix the ratio between thevelocities r = cP/cS = 2 (that is a very common choice in geophysicalapplications), the incidence angle h = p/4 and, for the DGSEM, wefix the parameter g = 2N2/h.

In Fig. 8 we show the grid dispersion errors with respect to thedegree N of the basis functions, fixing d = 0.2 (namely 5 gridpoints per wavelength). All the non-conforming approachesreproduce the same spectral convergence of the SEM. The SIPGand the MSEM reach the threshold value 10�13 for N = 6 whilethe NIPG for N = 9.

The grid dispersion as a function of sampling ratio d is shown inFigs. 9–12 for the degrees N = 2, . . . ,5, respectively. The aim of this

10−110−6

10−5

10−4

10−3

10−2

10−1

Sampling ratio δ

Grid

dis

pers

ion

(eS

)

N = 2

SIPGNIPGMSEMSEM

2

4

10−110−6

10−5

10−4

10−3

10−2

10−1

100

Sampling ratio δ

Grid

dis

pers

ion

(eP

)

N = 2

SIPGNIPGMSEMSEM

4

2

Fig. 9. Grid dispersion versus the sampling ratio d: N = 2.

10−110−10

10−8

10−6

10−4

10−2

Sampling ratio δ

Grid

dis

pers

ion

(eS

)

N = 3

SIPGNIPGMSEMSEM

4

6

10−110−10

10−8

10−6

10−4

10−2

Sampling ratio δ

Grid

dis

pers

ion

(eP

)

N = 3

SIPGNIPGMSEMSEM

4

6


2 4 6 8 1010−15

10−10

10−5

100

Degree N

Grid

Dis

pers

ion

(eS

)SEMSIPGNIPGMSEM

2 4 6 8 1010−15

10−10

10−5

100

Degree N

Grid

Dis

pers

ion

(eP

)

NIPGSIPGSEMMSEM

Fig. 8. Grid dispersion versus the polynomial degree N: d = 0.2 and incident angle h = p/4.


analysis is to establish a relation between the absolute value jeSj,resp. jePj, and the mesh size h (i.e., determine q, resp. q0, such thatjeSj ¼ OðhqÞ, resp. jePj ¼ Oðhq0 Þ). The order of convergence is esti-mated by the slope of these lines in Figs. 9–12.

From the results reported in Figs. 9–12 it seems that the SIPGconverges with order q ¼ q0 ¼ Oð2NÞ, as the SEM; whereas a

suboptimal order q ¼ q0 ¼ OðN þ 1Þ is observed for both NIPGand MSEM methods. These results are in agreement with [49].

Finally, in Figs. 13–16, we show the anisotropy (that is the ratiocS,h/cS and the ratio cP,h/cP) introduced by the numerical schemes.We consider N = 2, 3, 4 and five points per wavelength. For N > 4the anisotropy is very small for all the practical purposes. We

10−110−14

10−12

10−10

10−8

10−6

10−4

Sampling ratio δ

Grid

dis

pers

ion

(eS

)

N = 5

SIPGNIPGMSEMSEM

8

10

10−110−15

10−10

10−5

Sampling ratio δ

Grid

dis

pers

ion

(eP

)

N = 5

SIPGNIPGMSEMSEM

8

10


0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Fig. 13. Anisotropy curves cS,h/cS of the SEM (left) and MSEM (right): sampling ratio d = 0.2 for polynomial degrees N = 2 (–), N = 3 (–) and N = 4 (�-). For visualization purposes,the grid dispersion has been magnified by a factor 20.

10−110−14

10−12

10−10

10−8

10−6

10−4

10−2

Sampling ratio δ

Grid

dis

pers

ion

(eS)

N = 4

SIPGNIPGMSEMSEM

6

8

10−110−14

10−12

10−10

10−8

10−6

10−4

10−2

Sampling ratio δ

Grid

dis

pers

ion

(eP)

N = 4

SIPGNIPGMSEMSEM

8

6



notice that, for N = 2, in the SIPG and in the MSEM the waves areslightly delayed for all possible incident angles while in the NIPG

the waves are accelerated. In Tables 1 and 2 we also report themaximum value max0 6 h 6 2pjeSj and max0 6 h 6 2pjePj respectively.

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Fig. 15. Anisotropy curves cP,h/cP of the SEM (left) and MSEM (right): sampling ratio d = 0.2 for polynomial degrees N = 2 (--), N = 3 (–) and N = 4 (�-). For visualizationpurposes, the grid dispersion has been magnified by a factor 10.

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Fig. 14. Anisotropy curves cS,h/cS of the SIPG (left) and NIPG (right): sampling ratio d = 0.2 for polynomial degrees N = 2 (--), N = 3 (–) and N = 4 (�-). For visualization purposes,the grid dispersion has been magnified by a factor 20.


From these results it can be inferred that all the methods performin a very similar way.

6.4. Stability – DGSEM and MSEM

To derive the stability condition for the analyzed methods westart by considering the problem

cM €Uþ bKU ¼ 0; ð61Þ

where all the terms appearing in the above equation are defined onthe reference element XC (we omit the superscripts to ease thenotation). In the DG framework the matrices bK and cM are Aþ eBand M respectively, while in the Mortar approach they are equalto eQ > eA eQ and eQ >fM eQ respectively. Assuming that the solution isthe plane wave given in (51), substituting this expression in (61)and approximating the second order time derivative with theLeap-Frog scheme (32) or (33), we obtain the following eigenvalueproblem

bKU ¼ KcMU; ð62Þ

depending on the degrees of freedom inside the reference elementXC and where

K ¼ 4Dt2 sin2 xhDt

2

� :

In order to make explicit the dependence of K on both the meshsize h and the polynomial approximation degree N we rewrite(62) on bX ¼ ð�1;1Þ2. Collecting out the size of the elements it yieldstobKU ¼ K0cMU; ð63Þ

with K0 = (h/Dt)2sin2(xhDt/2). Defining the stability parameterq = cPDt/h, we deduce the relation

q2K0 ¼ c2P sin2 xhDt

2

� 6 c2

P ;

or equivalently

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

Fig. 16. Anisotropy curves cP,h/cP of the SIPG (left) and NIPG (right): sampling ratio d = 0.2 for polynomial degrees N = 2 (--), N = 3 (–) and N = 4 (�-). For visualization purposes,the grid dispersion has been magnified by a factor 10.

Table 1Maximum value max06h62pjeSj for N = 2, 3, 4.

N SEM SIPG NIPG MSEM

2 1.2684e�03 2.7156e�03 1.7540e�02 2.3728e�023 8.7429e�06 7.5949e�06 2.7196e�04 2.2247e�044 4.2894e�08 5.2818e�08 2.4372e�05 1.3029e�06

Table 2Maximum value max06h62pjePj for N = 2, 3, 4.

N SEM SIPG NIPG MSEM

2 9.8805e�04 1.5402e�02 2.8146e�02 8.8991e�023 6.8628e�06 1.8732e�05 5.6250e�04 2.4218e�044 3.1687e�08 3.3492e�07 6.8619e�05 1.5325e�06


q 6 cP1ffiffiffiffiffiK0p ¼ ccflðK0Þ; ð64Þ

As noted in [17], ccfl is a function of K0 and then depends implicitlyon the wave vector j through the matrices bK and cM. Moreover,inequality (64) must be fulfilled for all the eigenvalues and all thewave vectors j = 2pd/h(cos(h), sin(h)). Thus, the stability conditionis given by

q ¼ min16j6m

min06h62p

ccflðK0jðhÞÞ; ð65Þ

where h is the incident angle of the plane wave and m is the numberof the eigenvalues of problem (63). We remark that condition (65) isequivalent to requiring that

q 6cðk;lÞffiffiffiffiffiffiffiffiffiffiffi

Kmaxp ;

where Kmax is the largest eigenvalue of problem (63) and c(k,l) is apositive constant. Thus, by estimating Kmax in terms of h and N, it ispossible to determine a bound for q.

In the DG approach, the bilinear form bKð�; �Þ associated to thematrix bK in (62) takes the formbKðu;vÞ ¼ Z

XC

rðuÞ : eðvÞ �X

f¼fT;B;R;Lg

Zcf

rðuÞ : v n

þ hZ

cf

ðu� gf Þ n : rðvÞ þ gf

Zcf

ðu� gf Þ n : v n;

where the functions u;v 2 VDGd are zero outside XC, and n is the nor-

mal unit vector pointing outside XC. According to the plane wavehypothesis made at the beginning of Section 6, we take

gf ¼ ebf u; for f ¼ fT;B; L;Rg:

Following [50], it is easy to prove that

bKðu;uÞ 6 cðk;l;aÞN4

h2 kuk20;XC

:

Thus, for the generalized eigenvalue problem (63), we can derivethe estimate

Kmax 6 cðk;l;aÞN4

h2 ;

and consequently

K0max 6 cðk;l;aÞN4: ð66Þ

For the MSEM we observe that writing K0max by the generalized Ray-leigh quotient yields

K0max ¼ supv2R2mnf0g

ðbKv;vÞXC

ðcMv;vÞXC

¼ supv2R2mnf0g

ð eQ >eA eQ v;vÞXC

ð eQ >fM eQ v;vÞXC

¼ supv2R2mnf0g

ðeA eQ v; eQ vÞXC

ðfM eQ v; eQ vÞXC

¼ supw¼eQ v2R2nnf0g9 i¼1;...;2m:v�ei–0

ðeAw;wÞXC

ðfMw;wÞXC

6 supv2R2nnf0g

ðeAv;vÞXC

ðfMv;vÞXC

¼ supv2R2nnf0g

ðAv;vÞXC

ðMv;vÞXC

; ð67Þ

where m = (N2 + 3) and n = (N + 1)2. In this way we obtain an upperbound for the maximum eigenvalue of (63) when using MSEMapproximation. In fact the last term in (67) is exactly the maximumeigenvalue of the SEM discretization for which the following esti-mate holds (cf. [51])

c1N46 K0max 6 c2N4;

for c1 and c2 positive constants. We remark that in agreement with[50], we notice that for d = 2 the estimate (66) does not depend on h.This behaviour is confirmed from the results in Table 4. Finally, wecan resume the stability analysis in the following statement.

Table 3Computed upper bound for the stability parameter q using r = 1.414: rate of decaywith respect to N.

N SEM MSEM SIPG NIPG

2 0.3376 0.3333 0.2621 0.21633 0.1967 0.1770 0.1368 0.10454 0.1206 0.1118 0.0795 0.06075 0.0827 0.0776 0.0530 0.04006 0.0596 0.0570 0.0374 0.02817 0.0449 0.0434 0.0280 0.02108 0.0351 0.0342 0.0216 0.01629 0.0281 0.0277 0.0172 0.012910 0.0231 0.0227 0.0140 0.0105N-rate �1.8463 �1.8253 �1.9247 �1.9360

Table 4Computed upper bound for the stability parameter q0 using r = 1.414. Note that q0 isproportional to qN2 thus constant for different choices of Dx.

N SEM MSEM SIPG NIPG

2 0.6752 0.6667 0.5241 0.43263 0.7115 0.6403 0.4951 0.37824 0.6983 0.6474 0.4607 0.35165 0.7039 0.6608 0.4515 0.34096 0.7017 0.6712 0.4400 0.33157 0.7009 0.6769 0.4360 0.32738 0.7005 0.6819 0.4303 0.32289 0.6994 0.6878 0.4282 0.3206

10 0.6995 0.6871 0.4247 0.3180


Proposition 1. For every l > 0, k P 0 and a P amin > 0, the CFLcondition (64) is satisfied for both MSEM and DGSEM if there exists apositive constant c⁄(k,l,a) such that

q 6c�ðk;l;aÞ

N2 : ð68Þ

Moreover for the MSEM and the NIPG it holds amin = 0 andc⁄(k,l,a) = c⁄(k,l).

We remark that for the SIPG, the constant c⁄(k,l,a) is propor-tional to a�1/2 (cf. [50]), then a less restrictive bound for q in (68)is achieved when a = amin. Moreover it is possible to determine ex-actly the threshold value amin (cf. [52] for the elliptic case), but thisis not the objective of this study. For the following numerical sim-ulations we choose a = 1.

6.5. Stability – numerical results

To determine an upper bound for the stability parameter q wefix d = 0.2 and the ratio r = 1.414. This choice gives a more restric-tive stability condition: higher values of r = cP/cS produce milderstability condition [17]. As for the grid dispersion analysis we havefixed g = 2N2/h for the SIPG and the NIPG methods. In Table 3 areshown the estimated threshold values for q, for N = 2, . . . ,10. Theconstants for the SIPG are around 70% with respect the SEM, whilefor the MSEM are around 95%. The NIPG has constants always morerestrictive than those of SIPG.

In Table 3 it is also shown the asymptotic behaviour of the ccfl

with respect to N (N-rate): as expected the decay rate of q isapproximately proportional to N�2. We remark that the N-rate iscomputed using polynomial degree up to 20. In Fig. 17 we showthe trend of Kmax with respect to the polynomial degree, in agree-

2 4 6 8 10 12 14 16 1820100

101

102

103

104

105

106

N

Λ’ m

ax

SEMMSEMSIPGNIPG

4

Fig. 17. Kmax versus the polynomial degree N for the generalized eigenvalueproblem (63).

ment with the theoretical estimate. In practice, the time step is of-ten bounded, not by the size of the spectral elements h, but by thesmaller space increment Dx, then, in Table 4 we compute the upperbounds for the modified stability parameter q0 = cPDt/Dx. It is evi-dent that the CFL condition (65) is less restrictive for the MSEMthan for the DGSEM. Then the MSEM allows for larger time stepDt in the explicit time integration scheme.

7. Accuracy and order of convergence

Firstly we discuss the accuracy of the MSEM and of the DGSEMon a test case where the exact solution is known. We analyze awave propagation problem in X = (0,1)2, setting the elastic param-eters k = l = q = 1, and choosing f such that the exact solution of (1)is

uðt; x; yÞ ¼ sinffiffiffi2p

pt� � � sin2ðpxÞ sinð2pyÞ

sinð2pxÞ sin2ðpyÞ

" #: ð69Þ

The Dirichlet boundary conditions on oX and the initial displace-ment u0 and initial velocity u1 are set accordingly.

We then subdivide X into two subregions X1 and X2 with par-titioning T h1 and T h2 and fix N1 and N2 as the degree of the spectralexpansion in each subregion respectively. The skeleton is definedby S ¼ @X1 \ @X2 as it is shown in Fig. 18. In order to study theproperty of convergence of MSEM an DGSEM with respect toh = (h1,h2) and N = (N1,N2) we examine two different situations:the first corresponding to a Cartesian matching grid (Fig. 18, left)while the second to a Cartesian non-matching grid (Fig. 18, right),referred to as grid A and grid B, respectively. In Fig. 18 is shownthe first level (L1) of refinement for both grids, corresponding tothe initial mesh sizes h1 and h2 for X1 and X2. At each further stepof refinement (for a maximum number of four steps), we consider auniform refinement of the grids at the previous level, in particularfor grid A, Li refers to h1 = h2 = 2�i whereas for grid B, Li refers toh1 = 2�i and h2 (2/3)h1. For the time integration we employ thesecond order explicit Leap-Frog scheme described in Section 4.3.

For SEM approximations we recall that, under suitable assump-tions on the partition size h and on the polynomial degree N an apriori error bound of the following form holds (see [14])

ku� udk0 6 C Dt2 þXK

k¼1

h2rkk

N2skk

kuk2Hsk ðXkÞ

( )12

24 35;for C positive constant. Here sk represents the Sobolev regularity ofu in Xk, rk = min(Nk + 1,sk) and Dt the time step. Similar bounds canbe proved for DGSEM and MSEM on the basis of the estimates inTheorem 3 in Section 5.

In particular, if the mesh size h is constant (i.e., h1 = � � � = hK = h)we expect exponential convergence in N = minkNk, whereas if thespectral order of approximation N is fixed (i.e., N1 = � � � = NK = N)we expect algebraic convergence in h = maxkhk.

Fig. 18. First level of refinement (L1) for the grid A (left) and B (right). The end points of the skeleton S are highlighted by two circles.


In the family of proposed DGSEMs, we analyze in detail the SIPGmethod (i.e., h = �1 in (14)) because it exhibits better perfor-mances in term of grid dispersion and stability (see Section 6).

In Fig. 19 (resp. Fig. 21) we report the L2-error using MSEM andSIPG with grid A for different choices of N (resp. d.o.f.). The esti-

2 4 6 8 1010−8

10−6

10−4

10−2

N

|| u

− u

δ|| L2 ( Ω)

SEMMSEM N2 = N1MSEM N2 = N1+1MSEM N2 = N1+2

Fig. 19. Computed errors versus the polynomial degree N: MSEM (left) and DGSEM (rightgrid A and the refinement level L2.

2 4 6 8 1010−10

10−8

10−6

10−4

10−2

N

|| u

− u δ

|| L2 (Ω)


Fig. 20. As in Fig. 19

mated norm is computed at the time t⁄ = 2 using Dt = 5 � 10�4. Allplots in Figs. 19–22 are displayed in semilogarithmic scale.

The results show that both methods have the same rate ofconvergence as the SEM one. In Fig. 20 (resp. Fig. 22) it is shown

2 4 6 8 1010−8

10−6

10−4

10−2

N

|| u

− u

δ|| L2 ( Ω)

SEMSIPG N2 = N1SIPG N2 = N1+1SIPG N2 = N1+2

) at the observation time t⁄ = 2 using Dt = 5 � 10�4. The results are obtained with the

2 4 6 8 1010−10

10−8

10−6

10−4

10−2

N

|| u

− u

δ|| L2 ( Ω)

SEMSIPG N2 = N1 SIPG N2 = N1+1SIPG N2 = N1+2

with Dt = 10�4.

10−110010−5

10−4

10−3

10−2

h1

|| u

− u δ|| L2 (Ω

) 3

MSEM AMSEM BSIPG ASIPG BSEM

10−1100

10−8

10−6

10−4

h1

|| u

− u δ|| L2 (Ω

)

5


Fig. 23. Computed errors versus the mesh size: N1 = N2 = 2, Dt = 10�3 (left) and N1 = N2 = 4 and Dt = 10�4 (right). The error in the L2-norm is computed at the observation timet⁄ = 2 for all the refinement levels L1–L4. The suffixes A, B in the legend refer to the grids employed in the computation.

0 1000 2000 3000 4000 500010−10

10−8

10−6

10−4

10−2

dof

|| u

− u

δ|| L2 (Ω

)


0 1000 2000 3000 4000 500010−10

10−8

10−6

10−4

10−2

dof

|| u

− u

δ|| L2 (Ω

)


Fig. 22. As in Fig. 21 with Dt = 10�4.

0 1000 2000 3000 4000 500010−8

10−7

10−6

10−5

10−4

10−3

10−2

dof

|| u

− u

δ|| L2 (Ω)


0 1000 2000 3000 4000 500010−8

10−7

10−6

10−5

10−4

10−3

10−2

dof

|| u

− u

δ|| L2 (Ω)


Fig. 21. Computed errors versus the number of dof: MSEM (left) and DGSEM (right) at the observation time t⁄ = 2 using Dt = 5 � 10�4. The results are obtained with the grid Aand the refinement level L2.


the L2-error using a different time step Dt = 10�4 for differentchoices of N (resp. d.o.f.).

The results confirm that MSEM and SIPG have both exponentialconvergence in N, until the threshold value given byDt2 is reached.

Now, we fix N1 = N2 and we study the accuracy of the two meth-ods with respect to the mesh size h. For each level of refinement we

compute the error in L2-norm obtained using grid A and grid B.The algebraic order of convergence OðhNþ1Þ is achieved in bothcases for different choices of N and Dt (see Figs. 23 and 24).

Finally in Figs. 25 and 26 we show a qualitative analysis of sta-bility of MSEM and SIPG applied to this test case. The results are inagreement with those obtained in Section 6 and confirm that the

0.02 0.04 0.06 0.08 0.1 0.12 0.14

10−4

10−3

10−2

10−1

100

101

Δt

|| u

− u δ|| L2 (Ω

)

L1L2

L3

L4

0.02 0.04 0.06 0.08 0.1 0.12 0.14

10−4

10−3

10−2

10−1

100

101

Δt

|| u

− u δ|| L2 (Ω

)

L4

L3

L2 L1

Fig. 25. Stability analysis of MSEM (left) and SIPG (right) with respect to the mesh size h. The L2-error is computed at the observation time t⁄ = 20 with N1 = N2 = 2. Solid lines(–) correspond to SEM approximations, while dashed lines (--) to non-conforming approximations: MSEM on the left and SIPG on the right.

0.02 0.04 0.06 0.08 0.1 0.12 0.1410−6

10−5

10−4

10−3

10−2

10−1

100

101

Δt

|| u

− u δ|| L2 (Ω

)

N=2N=4

N=6

0.02 0.04 0.06 0.08 0.1 0.12 0.1410−6

10−5

10−4

10−3

10−2

10−1

100

101

Δt

|| u

− u δ|| L2 (Ω

)

N=2N=4

N=6

Fig. 26. Stability analysis of MSEM (left) and SIPG (right) with respect to the polynomial order N. The L2-error is computed at the observation time t⁄ = 20 for the refinementlevel L1 using the grid A. Solid lines (–) correspond to SEM approximations, while dashed lines (--) to non-conforming approximations: MSEM on the left and SIPG on theright.

101 102 103 10410−5

10−4

10−3

10−2

10−1

dof

|| u

− u δ

|| L2 (Ω)


102 103 10410−9

10−8

10−7

10−6

10−5

10−4

10−3

dof

|| u

− u

δ|| L2 (Ω)


Fig. 24. Computed errors versus the mesh size: N1 = N2 = 2, Dt = 10�3 (left) and N1 = N2 = 4 and Dt = 10�4 (right). The error in the L2-norm is computed at the observation timet⁄ = 2 for all the refinement levels L1–L4. The suffixes A, B in the legend refer to the grids employed in the computation.


region of stability for MSEM is larger than that for SIPG. Then, forexplicit time integration scheme, MSEM is preferable to SIPG.

8. An application of geophysical interest

In this section we analyze the seismic response of an alluvial ba-sin. We consider the viscoelastic model (5) in the computational

domain (x,z) 2X = (0,2 � 104 m) � (�9.6 � 102 m, f(x)) where f de-scribes the top profile of the valley, see Fig. 27. The bottom andthe lateral boundaries are set far enough from the point sourceso to avoid any interference of possible reflections from non-perfectly absorbing boundaries with the waves of interest thatare reflected, transmitted, or converted at the material or freesurfaces. We simulate a point source load of the form

Table 5Dynamic and mechanical parameters.

Layer cP (m/s) cS (m/s) q (kg/m3) f (1/s)

1 700 350 1900 0.031412 3500 1800 2200 0.06283


fðx; tÞ ¼ gðxÞhðtÞ;

where f is the external force introduced in (1). The function g de-scribes the space distribution of the source and is written in theform

gðxÞ ¼ dðx� xSÞw;

where d represents the Dirac distribution, xS the source location andw the direction of the applied force (cf. [27]). Alternative source dis-tributions can be expressed in terms of gradient or curl of suitablepotential functions, giving rise to pure pressure and shear waves:more complex and realistic source mechanisms are based on tenso-rial models (cf. [1]). The source time history is given by a Ricker-type function with maximum frequency mmax = 3 Hz, defined as

hðtÞ ¼ h0½1� 2bðt � t0Þ2� exp½�bðt � t0Þ2�; ð70Þ

where h0 is a scale factor, t0 = 2 s is the time shift andb ¼ p2m2

max ¼ 9:8696 s�1 is a parameter that determines the widthof the wavelet (70). A significant property is the cut-off at bothlow and high frequencies: the spectrum of the signal is maximumat mp ¼

ffiffiffibp

=pHz and is practically negligible for frequencies higherthan mco = 3mp.

In Figs. 27 and 28, we show the two different mesh configura-tions. Fig. 27 shows a regular, structured grid with a mesh spacingof h 40 m. The mesh size is chosen small enough to describe with

Fig. 28. Non-conforming, quasi-structured grid with overlap with a mesh spacing h1 4on the top of the valley and point source xS within the bedrock. Bottom: zoom of the va

Fig. 27. Conforming, structured grid with a mesh spacing of h 40 m at the interface besource xS within the bedrock. Bottom: zoom of the valley profile.

sufficient precision the physical profile of the valley. Fig. 28 showsan irregular, quasi-structured grid with overlap with a mesh spac-ing h1 40 m for layer 1 (basin) and h2 120 m for layer 2 (bed-rock). The finest mesh is used to describe the physical boundaryof the valley while the coarsest mesh the bedrock. This type ofoverlapping discretizations are handled by the Mortar techniquedescribed in Section 3.2.

We assign constant material properties within each region asdescribed in Table 5.

This regular conforming grid in Fig. 27 is used with SEM discret-ization to produce a reference solution for the problem and pro-vides a sufficiently accurate discretization, since further meshrefinements generates quasi-identical seismograms.

In Figs. 29 and 30 we compare the horizontal and vertical dis-placement recorded by receiver R1 placed on the free surface ofthe valley (cf. Figs. 27 and 28).

0 m for layer 1 (basin) and h2 120 m for layer 2 (bedrock). Top: receiver R1 placedlley profile.

tween the two materials. Top: receiver R1 placed on the top of the valley and point

0 5 10 15 20 25 30 35 40 45 50−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time [s]

Dis

plac

emen

t [m

]

MSEMSEM

Fig. 29. Horizontal displacement recorded by the receiver R1 on the free surface for the valley. Comparison between SEM and MSEM, N = 4.

0 5 10 15 20 25 30 35 40 45 50−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]

Dis

plac

emen

t [m

]

MSEMSEM

Fig. 30. Vertical displacement recorded by the receiver R1 on the free surface for the valley. Comparison between SEM and MSEM, N = 4.

Fig. 31. Time histories of the receivers on the top of the surface obtained with MSEM (N = 4).


The high discontinuities between the mechanical properties ofthe materials produce high oscillations and perturbations on the

wave front. All these complex phenomena are well captured byboth SEM and MSEM using fourth order spectral elements. We


remark that with MSEM we reduce the computational effort for thegeneration of the grid as well as the problem complexity (from61,385 spectral nodes with SEM to 48,091 spectral nodes withMSEM). In Fig. 31 we show the time histories of the seismogramsrecorded by some receivers on the free surface of the domain, ob-tained using MSEM. It can be observed that the wave which startstravelling from the point source remains trapped into the valley,where it is amplified and where phenomena of reflection andrefraction arise. This phenomenon is relevant in some geophysicalcontexts, e.g. it has occurred in the Gubbio valley (in Italy) on theoccasion of the earthquake of September 27, 1997. We refer to [53]for a detailed analysis.

9. Error analysis

In this section we report the proofs of the theorems stated inSection 5.

9.1. Semi-discrete error analysis – DGSEM

We recall from [54] the following approximation result. For anyXk � X; cj 2 F I and u 2 Hsk ðXkÞ there exists a sequenceuI 2 Q Nk

ðXkÞ, for Nk = 1,2, . . . , such that

ku� uIkq;Xk6 C

hmk�qk

Nsk�qk

kuksk ;Xk; 0 6 q 6 sk; ð71Þ

ku� uIk0;cj6 C

hmk�1=2k

Nsk�1=2k

kuksk ;Xk; sk > 1=2; ð72Þ

where mk = min(Nk + 1,sk) and C is a positive constant independentof hk and Nk. For further use we also introduce the projection oper-ator P :V ? Vd such that

ADGðPu;vÞ ¼ ADGðu;vÞ 8v 2 Vd: ð73Þ

Notice that Pu is well defined thanks to Lemma 1. Moreover, sincePu is a projection, we clearly have

jjju�Pujjj2DG 6 2jjju� uI jjj2DG þ 2jjjuI �Pujjj2DG

6 2jjju� uI jjj2DG þ2MjjjjuI � ujjj2DGjjju�Pujjj2DG

6 2jjju� uI jjj2DG

þ Mj�jjju� uI jjj2DG þ

M�jjjju�Pujjj2DG;

for any positive constant �. Therefore for � = j/2M we have

jjju�Pujjj2DG 6 Cjjju� uI jjj2DG:

Finally, by using (71) and (72), it is easy to see that Pu satisfies thefollowing approximation property.

Lemma 3. There exists a positive constant C such that

jjju�PujjjDG 6 CXK

k¼1

h2mk�2k

N2sk�3k

kuk2sk ;Xk

!1=2

: ð74Þ

Moreover it holds

ku�Puk0 6 CXK

k¼1

h2mkk

N2sk�2k

kuk2sk ;Xk

!1=2

: ð75Þ


Proof. We start showing inequality (74). Let v ¼ u� uI . Using theinverse-trace inequality and the interpolation estimates (71) and(72) it holds

jjjvjjj2DG ¼ kvk2DGþ

XK

k¼1

hk

N2k

!2

jvj22;Xk

6 CXK

k¼1

krvk20;XkþXM

j¼1

gcjksvtk2

0;cjþXK

k¼1

hk

N2k

!2

jvj22;Xk

6 CXK

k¼1

hmk�2k

Nsk�2k

kuk2sk ;XkþXK

k¼1

h2mk�2k

N2sk�3k

kuk2sk ;Xk

XK

k¼1

h2mk�2k

N2skk

kuk2sk ;Xk

" #

6 CXK

k¼1

h2mk�2k

N2sk�3k

kuk2sk ;Xk

;

where Nk P 1, mk = min(Nk + 1,sk). Next, we show (75). We setv = Pu � u. We assume that X is sufficiently smooth so that thesolution of the dual problem

�r � rðUÞ ¼ v; in X;

U ¼ 0; on @X;

�belongs to H2(X), with continuous dependence on v, i.e., $C > 0:

kUk2 6 Ckvk0: ð76Þ

Integrating by parts element wise yields:

kvk20 ¼ �r � rðUÞ;vð ÞX ¼

XK

k¼1

rðUÞ;rvð ÞXk� rðUÞ � n;vð Þ@Xkn@X

h i:

Thanks to the symmetry of r and (12) we obtain

kvk20 ¼

XK

k¼1

rðUÞ : eðvÞð ÞXk�XM

j¼1

frðUÞg; svtð Þcj;

since thanks to the regularity of U, sr(U)t = 0 on each cj. By sub-tracting the orthogonality equation for any U� 2 Vd : ADGðv;U�Þ ¼0, using the symmetry of r and the regularity of U we have

kvk20 ¼

XK

k¼1

rðU�U�Þ : eðvÞð ÞXk� ð1þ hÞ

XM

j¼1

frðUÞg; svtð Þcj

þ hXM

j¼1

frðU�U�Þg; svtð ÞcjþXM

j¼1

frðvÞg; sU�U�tð Þcj

�XM

j¼1

gcjsrðvÞt; sU�U�tð Þcj

:

By using the inverse and trace inequality, the estimate (76) and theapproximation property (74) we have

kvk20;Xk6 C

hk

N1=2k

kUk2;XkjjjvjjjDG 6 C

hk

N1=2k

kvk0;XkjjjvjjjDG:

The inequality (75) follows now using (74). h

Now, for all 0 6 t 6 T we set u�DG ¼ u�DGðtÞ the unique solution inVd of the problem

dtt qu�DG;v� �

þA�DGðu�DG;vÞ ¼ LðvÞ; 8v 2 Vd: ð77Þ

From Lemma 1 and standard techniques, follows that the varia-tional problem in (77) is well posed. From the results given in[12,15], the estimates (74) and (75), we have

Lemma 4. There exists a positive constant C such that for all t 2 [0,T]

jjjðu� u�DGÞðtÞjjjDG 6 CXK

k¼1

h2mk�2k

N2sk�3k

kuk2H2ð0;t;Hsk ðXkÞÞ

( )1=2

; ð78Þ


:


Proof. Let Pu be defined as in (73). By the triangle inequality wehave

jjju� u�DGjjjDG 6 jjju�PujjjDG þ jjjPu� u�DGjjjDG ¼ T1 þ T2:

Estimate (74) yields

jT1j 6 CXK

k¼1

h2mk�2k

N2sk�3k

kuk2sk ;Xk

!1=2

: ð79Þ

For the term T2 we set v ¼ u�DG �Pu and n = u � Pu. Denoting by/t ¼ @t/ the derivative of / with respect to time we have for t > 0

ðqvtt ;vÞ þ A�DGðv;vÞ ¼ ðqntt ;vÞ þ A�DGðn;vÞ 8v 2 L2ð0; T; VdÞ:

Denoting by Jðu;vÞ ¼PM

j¼1gcjðsut; svtÞcj

and recalling (73), theabove equation reduces to

ðqvtt ;vÞ þ A�DGðv;vÞ ¼ ðqntt ;vÞ þ Jðnt ;vÞ 8v 2 L2ð0; T; VdÞ: ð80Þ

By choosing v = vt, the error equation (80) becomes

ðqvtt ;vtÞ þ ADGðv;vtÞ þ Jðvt;vtÞ ¼ ðqntt ;vtÞ þ Jðnt ;vtÞ:

We can rewrite the above equation as follows

12

dtkq1=2vtk20 þ

XK

k¼1

rðvÞ; eðvtÞ� �

Xkþ Jðv;vtÞ þ Jðvt;vtÞ

¼ ðqntt;vtÞ þ Jðnt;vtÞ þXM

j¼1

frðvÞg; svtt� �

cj

� hXM

j¼1

svt; frðvtÞg� �

cj

which is equivalent to

12

dtkq1=2vtk20 þ

12

dtkvk2DG þ Jðvt;vtÞ

¼ ðqntt;vtÞ þ Jðnt;vtÞ þXM

j¼1


cj� h

XM

j¼1

svt; frðvtÞg� �

cj

ð81Þ

Therefore, integrating (81) in time between 0 and t, noting that bydefinition v(0) = 0, we obtain:

12kq1=2vtðtÞk

20 þ

12kvðtÞk2

DG þZ t

0Jðvt ;vtÞ

¼Z t

0ðqntt;vtÞ þ

Z t

0Jðnt ;vtÞ � h

XM

j¼1

svt; frðvÞgð ÞcjðtÞ þ ð1

þ hÞZ t

0

XM

j¼1


cjþ 1

2kq1=2vtð0Þk

20: ð82Þ

We now bound each of the terms on the right-hand side of (82) thatinvolves integrals on cj, using the trace inequality:

ð1þ hÞZ t

0

XM

j¼1


cj

�� 6 C

2�

Z t

0kvk2

DG þ�2

Z t

0Jðvt ;vtÞ;

hXM

j¼1

svt; frðvÞgð Þcj

�� 6 �2 kvk2

DG þC

2�aJðv;vÞ;

8�; � > 0. We also haveZ t

0ðqntt;vtÞ 6

Z t

0

12kq1=2vtk

20 þ

Z t

0

12kq1=2nttk

20;

andZ t

0Jðnt;vtÞ 6

C2�

Z t

0Jðnt ; ntÞ þ

�2

Z t

0Jðvt ;vtÞ:

Then, Eq. (82) reduces to

12

q1=2vtðtÞ�� 2

0 þ12� �

2

� kvðtÞk2

DG þ ð1� �ÞZ t

0Jðvt ;vtÞ �

C2�a

Jðv;vÞ

612kq1=2vtð0Þk

20 þ

C2�

Z t

0kvk2

DG þZ t

0

12kq1=2vtk

20

þZ t

0

12kq1=2nttk

20 þ

C2�

Z t

0Jðnt ; ntÞ: ð83Þ

Taking � ¼ 1=4; � ¼ 1=2 and a P 4C we have

kq1=2vtðtÞk20 þ kvðtÞk

2DG 6 C

Z t

0kq1=2vtk

20 þ kvk

2DG

� �þ kq1=2vtð0Þk

20

�þZ t

0kq1=2nttk

20 þ

Z t

0Jðnt; ntÞ

�:

By applying the Gronwall’s lemma [28] we obtain

kq1=2vtðtÞk20þkvðtÞk

2DG6C kq1=2vtð0Þk

20þZ t

0kq1=2nttk

20þZ t

0Jðnt;ntÞ

� �:

By the approximation property (74) it holds:

kq1=2vtð0Þk20 6 C

XK

k¼1

h2mkk

N2skk

kutk2L2ð0;t;Hsk ðXkÞÞ

;

Z t

0kq1=2nttk

20 6 C

XK

k¼1

h2mk�2k

N2sk�3k

kuttk2L2ð0;t;Hsk ðXkÞÞ

;

Z t

0Jðnt ; ntÞ 6 C

XK

k¼1

h2mk�2k

N2sk�3k


:

Therefore

jT2j 6 CXK

k¼1

h2mk�2k

N2sk�3k


þ kuttk2L2ð0;t;Hsk ðXkÞÞ

h i !1=2

: ð84Þ

Then, (78) follows by combining the estimate (79) and (84) and tak-ing the supremum over all t 2 [0,T]. h

Now, we are ready to prove Theorem 1.

Proof. Let u�DG 2 Vd be the solution of (77). From Lemma 1 we canshow that

jjjðuDG � u�DGÞðtÞjjjDG 6 Cjjjðu� u�DGÞðtÞjjjDG 8 t 2 ½0; T�: ð85Þ

In fact, it holds

jjjuDG � u�DGjjj2DG 6 CkuDG � u�DGk

2DG 6

CjADGðuDG � u�DG;uDG � u�DGÞ

¼ CjADGðu� u�DG;uDG � u�DGÞ

6CMjjjju� u�DGjjjDGjjjuDG � u�DGjjjDG:

Now, by the triangle inequality,

jjjðu� uDGÞðtÞjjjDG 6 jjjðu� u�DGÞðtÞjjjDG þ jjjðuDG � u�DGÞðtÞjjjDG;

then, the desired result is obtained using (78)–(85) and taking thesupremum over all t 2 [0,T]. h

9.2. Semi-discrete error analysis – MSEM

The crucial point of the MSEM error analysis relies on the con-struction of a modified elliptic projection operator P from V toVmortar

d satisfying optimal error estimate with respect to both hand N. In order to define it we need some preliminary approxima-tion results that we recall from [8,9,19].


For any non-mortar side c�n of S such that c�n ¼ c�nðkÞ � @Xk, wedefine the projection operator

pn : Ł2ðc�n Þh i2

! Kdðc�n Þ �2 \H1

0ðc�n Þ;

byZc�n

ðv � pnvÞ � bU ¼ 0 8 bU 2 Kdðc�n Þ �2

: ð86Þ

Then, see [9], for any non-negative real numbers s and q it holds

kv � pnvk�q;c�n6 C

hk

Nk

� qþs

kvks;c�n8v 2 Hsðc�n Þ:

We now define a lifting operator Rn : ½Kdðc�n Þ�2 \H1

0ðc�n Þ ! ½XdðXkÞ�2

such that Rnv ¼ v on c�n ; Rn vanishes on each side of Xk except onc�n and satisfies (see [55])

kRnvk1;Xk6 Ckvk1=2;c�n

8v 2 Kdðc�n Þ �2 \H1

0ðc�n Þ:

Moreover we introduce the operator P : V ! Vmortard defined by

Pu ¼XK

k¼1

uIjXkþX

C‘k�@Xk

.‘k

0@ 1A; ð87Þ

where uI satisfies (71) and (72) and

.lk ¼

0; if C‘k is a mortar edge;

Rn pnðu� uI Þjc�n

� �; if c�n ¼ C‘ðnÞ

kðnÞ is a non-mortar edge:

8><>:Finally we state the following approximation result (see [8] for theproof).

Lemma 5. There exists a positive constant C, independent of h and Nsuch that for any v 2 Hsk ðXkÞ it holds

kv � Pvk1;Xk6 C

hmk�1k

Nsk�1k

kvksk ;Xk; sk > 3=2; ð88Þ

with Nk P 1 and mk = min(Nk + 1,sk).We now define the modified elliptic projection P : V ! Vmortar

d

as:

AMðu�Pu;vÞ �XM

n¼1

rðuÞ : svtð Þc�n ¼ 0 8v 2 Vmortard : ð89Þ

Note that Pu 2 Vmortard is well defined since the bilinear form AMð�; �Þ

satisfies the coercive property (44).

Lemma 6. There exists a positive constant C such that

ku�Puk� 6 CXK

k¼1

h2mk�2k

N2sk�2k

kuk2sk ;Xk

!1=2

sk > 3=2: ð90Þ

with Nk P 1 and mk = min(Nk + 1,sk). Moreover it holds

ku�Puk0 6 CXK

k¼1

h2mkk

N2skk

kuk2sk ;Xk

!1=2

: ð91Þ

Proof. Using the projection operator introduced in (87) we rewritethe Eq. (89) obtaining

AMðPu�Pu;vÞ ¼ �AMðu� Pu;vÞ þXM

n¼1

rðuÞ : svtð Þc�n ¼ 0

8v 2 Vmortard :

Choosing v ¼ Pu�Pu and using the boundedness and coerciveproperty (43) and (44) respectively, we have

jkvk2� 6 Mku� Puk�kvk� þ

XM

n¼1

j rðuÞ : svtð Þc�n j:

Now, from [9, Proposition 3.1] we have

Xn

rðuÞ : svtð Þc�n�� 6 C

XK

k¼1

hmk�1k

Nsk�1k

kuksk ;Xkkvk�: ð92Þ

Therefore,

kvk� 6 C ku� Puk� þXK

k¼1

hmk�1k

Nsk�1k

kuksk ;Xk

" #:

(90) is obtained combining the above inequality with the estimate(88) and using the triangle inequality

ku�Puk� 6 ku� Puk� þ kPu�Puk�:

To prove (91) we set v = Pu � u. By duality arguments (see proof ofLemma 3) and integrating by parts on each element yields:

kvk20 ¼ �r � rðUÞ;vð ÞX ¼ AMðU;vÞ �

XM

n¼1

rðUÞ : svtð Þc�n ;

or equivalently,

kvk20 ¼ AMðU� PU;vÞ þ AMðPU;vÞ �

XM

n¼1

rðUÞ : svtð Þc�n :

Using the symmetry of r and the properties (89) and (86) we have

kvk20 ¼ AMðU� PU;vÞ þ

Xn

rðuÞ : sPUtð Þc�n

�XM

n¼1

ðrðUÞ � pnrðUÞÞ : svtð Þc�n :

We now bound the three terms on the right-hand side of the aboveequation. Using (43) we have

jAMðU� PU;vÞXkj 6 C

hk

Nkkvk0kvk1;Xk

for any Xk, k = 1, . . . ,K. By trace inequality we obtain

rðuÞ : sPUtð Þc�n�� 6 C

hmkk

Nskk

kvk0kuksk ;Xk;

ðrðUÞ � pnrðUÞÞ : svtð Þc�n�� 6 C

hk

Nkkvk0kvk1;Xk

;

for any c�n � @Xk. Therefore we conclude combining the above esti-mates with (90). h

Let uM = uM(t) be the solution in Vmortard of the variational prob-

lem (45). From the results given in [8,19], the estimate (90) and(91) we have the following

Lemma 7. There exists a positive constant C such that for all t 2 [0,T]it holds

kðPu� uMÞðtÞk� 6 CXK

k¼1

h2mk�2k

N2sk�2k


( )1=2

; ð93Þ


Proof. We introduce the modified elliptic projection Pu as in (89)and we set n = u �Pu and v = uM �Pu. When multiplying the firstline in (1) by a function v ¼ vðtÞ 2 L2ð0; T; Vmortar

d Þ and integratingby parts on each Xk, we observe that


q@ttu;vð Þ þ AMðu;vÞ þXM

n¼1

rðuÞ : svtð Þc�n ¼ LðvÞ:

Now subtracting (45) from the above equation we have, for anyv 2 L2ð0; T; Vmortar

d Þ,

q@ttðu� uMÞ;vð Þ þ AMðu� uM;vÞ þXM

n¼1

rðuÞ : svtð Þc�n ¼ 0;

or equivalently,

ðqvtt ;vÞ þ AMðv;vÞ ¼ ðqntt;vÞ þ AMðn;vÞ þXM

n¼1

rðuÞ : svtð Þc�n :

Choosing v = vt and using the property (89) we obtain

ðqvtt ;vtÞ þ AMðv;vtÞ ¼ qntt;vt

� �þ 2

XM

n¼1

rðuÞ : svtð Þc�n :

We rewrite it as follows

12

dtkq1=2vtk20 þ

12

dtAMðv;vÞ

¼ ðqntt;vtÞ þ 2XM

n¼1

dtðrðuÞ : svtð Þc�n � 2XM

n¼1

rðutÞ : svtt� �

c�n: ð94Þ

Therefore, integrating (94) in time between 0 and t, noting that bydefinition v(0) = 0 and using (44) we obtain12kq1=2vtðtÞk

20 þ

j2kvðtÞk2

�

6

Z t

0ðqntt ;vtÞ þ 2

XM

n¼1

rðuÞ : svtð Þc�n ðtÞ

� 2Z t

0

XM

n¼1

rðutÞ : svtt� �

c�nþ 1

2kq1=2vtð0Þk

20: ð95Þ

We now bound each of the terms in the right-hand side of (95) thatinvolves integrals on c�n , using the trace inequality:

2Z t

0

XM

n¼1

rðutÞ; svtð Þc�n

�� 6 C

XK

k¼1

h2mk�2k

N2sk�2k

Z t

0kutk2

sk ;Xkþ 1

2

Z t

0kvk2

� ;

2XM

n¼1

rðuÞ; svtð Þc�n

�� 6 C

2�

XK

k¼1

h2mk�2k

N2sk�2k

kuk2sk ;Xkþ �

2kvk2

� ;

"� > 0. We also haveZ t

0ðqntt;vtÞ 6

Z t

0

12kq1=2vtk

20 þ

Z t

0

12kq1=2nttk

20:

Then, inequality (95) yields

12kq1=2vtðtÞk

20 þ

j2� �

2

� kvðtÞk2

�

612

Z t

0kq1=2vtk

20 þ kvk

2�

� �þ 1

2

Z t

0kq1=2nttk

20

þ C2�

XK

k¼1

h2mk�2k

N2sk�2k

kuk2sk ;Xkþ 1

2kq1=2vtð0Þk

20

þ CXK

k¼1

h2mk�2k

N2sk�2k

Z t

0kutk2

sk ;Xk: ð96Þ

Choosing � such that j � � is bounded away from 0 we obtain


2�

6 CZ t

0kq1=2vtk

20 þ kvk

2�

� �þZ t

0kq1=2nttk

20

�þXK

k¼1

h2mk�2k

N2sk�2k

kuk2sk ;Xkþ kq1=2vtð0Þk

20 þ

XK

k¼1

h2mk�2k

N2sk�2k

Z t

0kutk2

sk ;Xk

#:

By applying the Gronwall’s lemma [28] it holds


2�

6 CZ t

0kq1=2nttk

20 þ

XK

k¼1

h2mk�2k

N2sk�2k

kuk2sk ;Xkþ kq1=2vtð0Þk

20

"

þXK

k¼1

h2mk�2k

N2sk�2k

Z t

0kutk2

sk ;Xk

#:

Using the approximation properties (90) and (91) it follows that:

kq1=2vtð0Þk20 6 C

XK

k¼1

h2mkk

N2skk


;

Z t

0kq1=2nttk

20 6 C

XK

k¼1

h2mk�2k

N2sk�2k

kuttk2L2ð0;t;Hsk ðXkÞÞ

:

Therefore we have

kvðtÞk2� 6 C

XK

k¼1

h2mk�2k

N2sk�2k

kuk2H2ð0;t;Hsk ðXkÞÞ

: �

Now, the proof of Theorem 2 is obtained using the triangle inequal-ity, estimates (90) and (91) and (93) and taking the supremum overall t 2 [0,T].

10. Conclusions

In this paper we proposed, analyzed and compared two differ-ent domain decomposition non-conforming high order numericaltechniques, namely the Mortar Spectral Element Method (MSEM)and the Discontinuous Galerkin Spectral Element Method(DGSEM), for the approximation of the elastic wave equation inheterogeneous media.

Both methods preserve the spectral accuracy typical of high or-der methods, allow geometrically non-conforming domain parti-tions where local meshes are independently generated and canhandle variable local polynomial degrees. Note that the subdomainpartition is constructed according to the information on the mate-rial properties.

Starting from a common weak formulation we described bothapproaches and highlighted their analogies and their differences.In particular, we gave special attention to the analysis of grid dis-persion, stability, and accuracy, which represent the main impor-tant features determining the reliability of a numerical method towave propagation problems. We numerically proved that theMSEM and the DGSEM do not suffer from grid dispersion. Indeedfive points per wavelength with spectral element approximationsof order four are sufficient to have negligible errors. For the stabilityanalysis we derived a precise CFL bound for the Leap-Frog schemewhen employed with the considered non-conforming approaches.The threshold values obtained for the DGSEM (resp. the MSEM)are around 70% (resp. 95%) of the ones typical of the Spectral Ele-ment Method (SEM). So, on the one hand, the symmetric versionof the DGSEM yields optimal error decays in the grid dispersionas occurs with the SEM. On the other hand, the MSEM allows largertime step in the time advancing scheme. Finally, both non-conform-ing techniques are well suited for parallel computations.

Acknowledgements

The first author was partially supported by Italian MIUR PRIN2008 grant ‘‘Analisi e sviluppo di metodi numerici avanzati per EDP’’.

The second and third authors acknowledge the financial sup-port of the Italian MIUR PRIN 2009 grant ‘‘Modelli numerici per ilcalcolo scientifico e applicazioni avanzate’’.


We thank Roberto Paolucci, Chiara Smerzini of the Departmentof Structural Engineering, Politecnico di Milano, and Marco Stu-pazzini of Munich RE for many interesting discussions, for provid-ing input for the geophysical application addressed in Section 8and their help in the analysis of numerical results.

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Non-conforming high order approximations of the elastodynamics equation

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