[Lecture Notes in Computational Science and Engineering] Advanced Computational Methods in Science...

Least-Squares Spectral Element Methods inComputational Fluid Dynamics

Marc Gerritsma and Bart De Maerschalck

Abstract The least-squares spectral element method (LSQSEM) is a relatively noveltechnique for the numerical approximation of the solution of partial differentialequations. The method combines the weak formulation based on the minimizationof a residual norm, the least-squares formulation, with the higher-order spectral el-ement discretization. A well-posed least-squares formulation leads to a symmetric,positive-definite system of algebraic equations which are highly amenable to well-established solvers such as the preconditioned conjugate gradient method. Further-more, the formulation is very robust in the sense that no stabilization operators arerequired to acquire convergent solutions. The spectral element discretization ren-ders high order accuracy to the scheme. This new numerical scheme is applied toincompressible, compressible and non-Newtonian flow problems.

1 Introduction

The least-squares spectral element method (LSQSEM) combines the weak formu-lation obtained from a least-squares minimization problem and a spectral elementdiscretization. In the early 70’s the least-squares technique was introduced for thesolution of partial differential equations by Bramble, [10, 11]. The method was al-ready described in Russian literature by Dziskariani, [30] and Lucka, [55]. In itsinitial form it was less competitive than the Galerkin formulation due to the highcondition numbers and the more stringent regularity requirements. It was only whenit was recognized that partial differential equations need to be rewritten in terms of

Marc GerritsmaDept. of Aerospace Engineering, TU Delft, Kluyverweg 1, 2629 HS Delft, The Netherlands e-mail:[email protected]

Bart De MaerschalckFlemish Institute for Technological Research, Boeretang 200, BE-2400 Mol, Belgium e-mail:[email protected]

Lecture Notes in Computational Science and Engineering 71, 179

DOI 10.1007/978-3-642-03344-5_7,

B. Koren and C. Vuik (eds.), Advanced Computational Methods in Science and Engineering,

© Springer-Verlag Berlin Heidelberg 2010


equivalent first order partial differential equations that the method gained renewedinterest.

In 2002/2003 two papers appeared shortly after another in the Journal of Com-putational Physics: [70] by Proot and Gerritsma, and [61] by Pontaza and Reddy. Inboth papers the least-squares formulation was combined with higher order spectralelement methods. Since then the number of practitioners and publications in thisfield has grown.

The outline of this chapter is as follows: In Section 2 the Rayleigh-Ritz prin-ciple, the Galerkin formulation and the least-squares formulation are presented. InSection 3 a brief overview of the spectral element methods is given. In Section 4convergence rates for well-posed problems are discussed. Applications of the nu-merical scheme are presented in the Sections 5 and 6, in which incompressible andcompressible flows are discussed, respectively. In Section 7 new developments arebriefly described. The final section, Section 8, gives an overview of the relevantliterature.

2 The least-squares formulation

Weak formulations of differential equations consist of functionals over functionspaces whose stationary points solve the original differential equation. These sta-tionary points can be distinguished in (global) minima or saddle points. If a partialdifferential equation can be rephrased in a minimization problem, very robust nu-merical methods can be developed. A well-known form is the Rayleigh-Ritz formu-lation.

If, on the other hand, no equivalent minimization formulation can be establisheda more general formulation in terms of stationary points can be devised. The disad-vantage of this approach is that the analysis of well-posedness of the weak formula-tion is much harder to perform and in many instances the well-posedness propertieson the continuous level are not inherited by a discrete approximation.

In order to position the least-squares formulation within the framework of weightedresidual methods and to introduce some notation and concepts we will briefly de-scribe the Rayleigh-Ritz method, the Galerkin formulation and eventually the least-squares formulation.

2.1 Rayleigh-Ritz method

Consider the Poisson problem

∆φ(x) = f (x) , x ∈Ω ⊂ Rd

, (1)

with

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Least-Squares Spectral Element Methods in Computational Fluid Dynamics

φ(x) = 0 for x ∈ Γ1 ⊂ ∂Ω and∂φ∂n

= 0 for x ∈ Γ2 ⊂ ∂Ω , Γ1∩Γ2 = /0 . (2)

This partial differential equation with homogeneous boundary conditions can beconverted in the following minimization problem:

Find φ ∈H which minimizes J (ψ) =

∫

Ω

[

12

(∇ψ)2 + f ψ]

dΩ . (3)

The set H contains all admissible functions for the minimization problem to makesense. Firstly, all functions in H must satisfy the essential boundary conditionψ(x) = 0 for x ∈ Γ1. Furthermore, the integral in (3) should exist. The set of func-

tions for which the gradient of a scalar is square integrable, i.e. ∇ψ ∈[

L2(Ω)]d

andψ ∈ L2(Ω), ensures that

J (ψ)≤C‖ψ‖2H , (4)

where H is defined as

H =

ψ |ψ ∈ L2(Ω) and ∇ψ ∈[

L2(Ω)]d

and ψ(x) = 0 for x ∈ Γ1

, (5)

with norm‖ψ‖2

H = ‖ψ‖2L2(Ω) +‖ψ‖2

[L2(Ω)]d , ∀ψ ∈ H . (6)

If φ minimizes the functional J (ψ) we need to have that

DJ (φ)ψ = 0 , ∀ψ ∈ H , (7)

where the linear operator DJ (ψ) is defined by, [79]

limε→0

J (ψ + εψ)−J (ψ)− εDJ (ψ)ψε

= 0 . (8)

For the functional given by (3) this gives

DJ (ψ)ψ = limε→0

∫

Ω

[

(∇ψ + ε∇ψ)2−2 f (ψ + εψ)− (∇ψ)2 + 2 f ψ]

dΩ

2ε(9)

= limε→0

∫

Ω

[

2ε∇ψ ∇ψ + ε2 (∇ψ)2−2ε f ψ]

dΩ

2ε(10)

=

∫

Ω[∇ψ ∇ψ− f ψ ] dΩ (11)

= 0 , ∀ψ ∈ H . (12)

This condition for a minimizer can be abstractly written as

a(ψ , ψ) = ( f , ψ) , ∀ψ ∈ H . (13)

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The bi-linear form a(·, ·) is in this case symmetric, i.e. a(u,v) = a(v,u).

Definition 1. A bi-linear form a(·, ·) on a normed linear space H is said to bebounded (or continuous) if ∃C < ∞ such that

|a(u,v)| ≤C‖u‖H‖v‖H , ∀u,v ∈H , (14)

and coercive on H if ∃α > 0 such that

a(u,u)≥ α‖u‖2H , ∀u ∈ H . (15)

⊓⊔

If a partial differential equation can be converted into a minimization problem overa function space H, where H is a Hilbert space and the bilinear symmetric forma(·, ·) is bounded and coercive, then the minimization possesses a unique solution.Furthermore, if the bilinear form is only coercive on a proper (closed) subspaceV ⊂ H, then the minimization problem possesses a unique solution in V .

This last property, the fact that one may look for a minimizer on a closed subspaceof H, allows one to construct finite-dimensional subspaces (finite elements) and lookfor an approximate solution in such a subspace V .

Furthermore, existence and uniqueness of a solution for the whole function spaceH are inherited by proper closed subsets V ⊂ H.

An alternative way to derive the conditions for a minimizer (12) is to take the dif-ferential equation and multiply it with an arbitrary function ψ ∈H. After integrationover the domain Ω this gives

∫

Ω(−∆φ + f ) ψ dΩ = 0 . (16)

Applying integration by parts and using the boundary conditions gives

0 =

∫

Ω(−∆φ + f ) ψ dΩ (17)

=

∫

Ω[−∇ · (ψ∇φ)+ ∇φ∇ψ + f ψ] dΩ (18)

=∫

Ω[∇φ∇ψ + f ψ ] dΩ −

∫

∂Ωψ (∇φ ,n) ,dΓ (19)

=

∫

Ω[∇φ∇ψ + f ψ ] dΩ , (20)

where n denotes the outward unit normal at the boundary ∂Ω . The boundary can bedecomposed as ∂Ω = Γ1 +Γ2. On Γ1 ψ = 0 and on Γ2 ∂φ/∂n = 0, so the boundaryintegral vanishes. The final result of the exercise shows that if φ solves the PDE, theweak formulation (20) is equal to the condition for a minimizer of the functionalJ , (12).

The variational approach described above was already used by Rayleigh in 1870,[74], Ritz in 1908, [75] and Galerkin in 1915, [32]. But it was only until the 1960’s

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that the full potential of solving differential equations by minimizing well-posedvariational statements was recognized in mechanical engineering.

2.2 Galerkin formulations

It is not always possible to convert a differential equation to an associated mini-mization problem. A well-known example with applications in fluid dynamics isthe steady linear advection equation given by

adudx

= 0 , x ∈ (0,1) , a > 0 , (21)

with boundary condition u(0)= u0. The method described by (16) is still applicable.This gives

a(u,v) =∫

Ωa

dudx

vdΩ = 0 . (22)

This formulation where the partial differential equation is weighted by a suitablychosen set of test functions is called the Galerkin formulation. This weak formu-lation does not correspond to a minimization problem and generally the space oftrial solutions, u, and test functions, v, are not the same. For the integral (22) tomake sense we need to have that u ∈H1(0,1) and v ∈ L2(0,1), so a setting in termsof Hilbert spaces is not possible. Consider the slightly more general case given by

Seek u ∈W such that: a(u,v) = f (v) , ∀v ∈V , (23)

where W and V are function spaces equipped with the norms ‖ · ‖W and ‖ · ‖V ,respectively. W is called the solution space and V is called the test space. We assumethat the bi-linear form is continuous as defined in Definition 1 and f ∈V ′. Then thefollowing theorem establishes the conditions for well-posedness, [31]

Theorem 1. Let W be a Banach space and let V be a reflexive Banach space. Leta ∈L (W ×V ;R) and f ∈V ′, then problem (23) is well-posed if and only if

∃α , infw∈W

supv∈V

a(w,v)‖w‖W‖v‖V

≥ α , (24)

∀v ∈V , (∀w ∈W , a(w,v) = 0) =⇒ (v = 0) . (25)

⊓⊔

This condition is referred to as the inf-sup condition, the Ladyzhenskaya-Babuska-Brezzi (LBB) condition or the Banach-Necas-Babuska condition. It is generallyquite hard for a given weak formulation to prove the existence of the positive con-stant α , the coercivity constant.

In contrast to the minimization problem, well-posedness on the continuous leveldoes not imply well-posedness for discrete conforming subspaces Wh ⊂W and Vh⊂

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V . This means that for every problem discrete function spaces need to be identifiedsuch that well-posedness also holds for (Wh,Vh). Different weak formulations thenrequire different approximating subspaces. Such may be the case with multi-physicssimulations such as fluid-structure interaction problems.

In order to establish well-posedness and to develop a consistent and convergentscheme a variety of stabilization operators are generally required.

2.3 Least-squares formulation

An alternative approach of establishing equivalence between differential equationsand weak formulations is offered by the so-called least-squares formulation. Theleast-squares formulation transforms partial differential equations into a minimiza-tion problem and in this sense extends the Rayleigh-Ritz formulation. The Rayleigh-Ritz formulation is limited to elliptic, self-adjoint problems. The previous subsec-tion, 2.1 and 2.2, demonstrated that it is favorable to have a weak formulation whichcorresponds to a minimization problem, but that it is not always possible to converta partial differential equation in terms of a minimization problem. The least-squaresformulation provides a framework in which partial differential equations can be castinto a minimization setting, thus retrieving the appealing properties of the Rayleigh-Ritz principle.

This is established by introducing a norm equivalence between the residual andthe error. Two norms, ‖ · ‖X and ‖ · ‖Y , are called equivalent when two positive con-stants, C1 and C2, exist such that

C1‖u‖X ≤ ‖u‖Y ≤C2‖u‖X , ∀u ∈ X . (26)

If we define the linear spaces to be

X = u |‖u‖X ≤ ∞ and Y = u |‖u‖Y ≤ ∞ , (27)

norm-equivalence states that both sets contain the same elements, X =Y . Equivalentnorms on a function space X define the same topology and more specifically, Cauchysequences in both norms are the same. For finite-dimensional spaces all norms areequivalent.

Let the abstract first order partial differential equation be given by

L u(x) = f (x) , ∀x ∈Ω , (28)

withRu(x) = g(x) , ∀x ∈ ∂Ω , (29)

where L is a first order linear partial differential operator and R is a linear traceoperator which prescribes boundary values. In the least-squares formulation, wenow look for function spaces, X and Y , with associated norms such that we have the

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norm-equivalence

α‖u‖X(Ω) ≤ ‖L u‖Y(Ω) +‖Ru‖Z(∂Ω) ≤C‖u‖X(Ω) , ∀u ∈ X . (30)

The space X is called the solution space and Y is called the residual space. Due tothe linearity of the differential and trace operator we have that, if uex ∈ X

α‖u−uex‖X(Ω) ≤ ‖L u− f‖Y(Ω) +‖Ru−g‖Z(∂Ω)≤C‖u−uex‖X(Ω) , ∀u ∈ X ,

(31)in which uex denotes the exact solution of (28-29). This equivalence tells us thatsmall residual norms correspond to small error norms and conversely, small errornorms correspond to small residual norms.

Based on this equivalence we can now define the so-called least-squares func-tional

J (u) =12

‖L u− f‖2Y(Ω) +‖Ru−g‖2

Z(∂Ω)

. (32)

Solving the abstract PDE (28-29) is now equivalent to minimizing the least-squaresfunctional J (u). So by applying the least-squares formulation, the problem hasbeen recast in terms of a minimization problem thus avoiding stability prerequisitessuch as inf-sup conditions as discussed in the previous section.

Minimization of the least-squares functional requires setting DJ (u)v = 0 for allv ∈ X analogous to the minimization procedure described in Section 2.1. Note alsothat the boundary conditions, Ru = g, are also incorporated in the minimizationprocess.

Without loss of generality, we can set g = 0 in the equation Ru = g. This al-lows us to incorporate the boundary conditions in the space X . When we do so, theboundary conditions are strongly enforced and can be dropped from the variationalstatement.

If Y is a Hilbert space this gives the weak formulation

(L u,L v)Y (Ω) = ( f ,L v)Y (Ω) , ∀v ∈ X . (33)

Note that this formulation is symmetric; interchanging u and v on the left hand sideof this equation leaves the expression unchanged. Furthermore, the basic existenceand uniqueness criterion, the norm-equivalence, (30), is inherited by conformingsubspaces.

Although in the previous discussion general function spaces X and Y weretreated, from a practical point of view it is convenient to choose function spaceswhich allow for easy calculation of the residual norm in the minimization process.So in many applications, the boundary conditions are already incorporated in thefunction space X and one takes Y (Ω) = L2(Ω). This leads to the least-squares func-tional

J (u) =12‖L u− f‖2

L2(Ω) . (34)

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Minimization is in the L2-sense, so point-wise convergence, in the L∞-sense, isonly attained if the exact solution is sufficiently regular. The next section discussesthe construction of conforming finite-dimensional subspaces of the solution spaceX .

3 Spectral element methods

Instead of seeking the minimizer over the infinite-dimensional space X we restrictour search to a conforming subspace Xh⊂X by performing a domain decompositionwhere the solution within each sub-domain is expanded with respect to a polynomialbasis. The domain Ω is sub-divided into K non-overlapping quadrilateral open sub-domains Ω k:

Ω =K⋃

k=1

Ω k, Ω k ∩Ω l = /0 , k 6= l . (35)

Each sub-domain is mapped onto the unit cube (−1,1)d , where d = dim(Ω). Withinthis unit cube the unknown function is approximated by polynomials. Generally aspectral element method based on Legendre polynomials, Lk(x) over the interval[−1,1], is employed, [15, 22, 52]. We define the Gauss-Lobatto-Legendre (GLL)nodes by the zeroes of the polynomial

(

1− x2)L′N(x) , (36)

and the Lagrange polynomials, hi(x), through these GLL-points, xi, by

hi(x) =1

N(N + 1)

(x2−1)L′N(x)LN(xi)(x− xi)

for i = 0, . . . ,N , (37)

where L′N(x) denotes the derivative of the Nth Legendre polynomial. For multi-dimensional problems tensor products of the one-dimensional basis functions areemployed in the expansion of the approximate solution. We can therefore expandthe approximate solution in each sub-domain in terms of a truncated series of theseLagrangian basis functions, which for d = 2 yields

uN(x,y) =N

∑i=0

N

∑j=0

ui jhi(x)h j(y), (38)

where the ui j’s are to be determined by the least-squares method. If we have con-verted a general higher order PDE to an equivalent first order system, C0-continuitysuffices to patch the solutions on the individual sub-domains together.

The integrals appearing in the least squares formulation, (34), are approximatedby Gauss-Lobatto quadrature

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∫ 1

−1f (x)dx ≈

P

∑i=0

f (xi)wi, (39)

where wi are the GL weights given by

wi =2

P(P + 1)

1

L2P(xi)

, i = 0, . . . ,P≥ N . (40)

It has been shown in [19] that it is beneficial for non-linear equations possessinglarge gradients to choose the integration order P higher than the approximation ofthe solution, N.

The method is not restricted to higher order methods based on Legendre poly-nomials. In [18] and [71] Lagrangian basis functions based on the Chebyshev poly-nomials were used for non-linear, time-dependent hyperbolic equations and the in-compressible Navier-Stokes equations, respectively. Other systems of orthogonalpolynomials can be easily introduced into the least-squares spectral element frame-work.

From a practical point of view, only least-squares formulations which allow forthe use of C0-finite or spectral elements are usable. Since C0-finite or spectral el-ement methods are based on piecewise continuously differentiable polynomials,standard finite and spectral elements can be used, which results in a very practi-cal method from an implementational point of view. This can be accomplished byfirst transforming the system into a first order system and subsequently requiringthat only (scaled) L2-norms are used in the quadratic least-squares functional, see(34). The transformation into a first order system has two important consequences.First of all, the continuity requirements between neighboring spectral elements aremitigated such that C0-finite or spectral elements can be used (in case the residualsare measured by L2-norms). Secondly, the transformation will keep the conditionnumber of the resulting discrete system under control [3, 14, 16].

4 Convergence and a priori error estimates

The H1− and L2-spaces are particularly suitable as the function spaces X and Y inleast-squares finite or spectral element methods resulting from the minimization offirst order partial differential equations with strongly imposed boundary conditions.To appreciate this, assume that we have the following norm-equivalence

α ‖u‖H1(Ω ≤‖L u‖L2(Ω)≤C‖u‖H1(Ω) , ∀u∈X =

u ∈ H1(Ω) |Ru = 0 on ∂Ω

,(41)

where the space X represents the space of functions which already satisfy the homo-geneous boundary condition and for which the function itself and its first derivativesare square integrable over the domain Ω . Based on the norm-equivalence (41), thequadratic least-squares functional can be obtained which upon minimization yieldsthe weak formulation of the least-squares problem. If one uses a conforming finite-

187


dimensional subspace Xh ⊂ X = H1(Ω), then one can approximate uh ∈ Xh bypiecewise continuously differentiable polynomials. In [68], it has been shown thatleast-squares methods based on the norm-equivalence (41) yield the following errorestimate:

∥

∥

∥u−uh

∥

∥

∥

1≤ C inf

vh∈Xh

∥

∥

∥u− vh

∥

∥

∥

1, (42)

where the constant C is given by C = 1+2C2/α2, with the constants α and C from

(41). Here the subscript 1 in the norm ‖ · ‖1 refers to the Sobolev norm ‖ · ‖H1(Ω),while ‖ · ‖0 will denote ‖ · ‖H0(Ω) = ‖ · ‖L2(Ω).

Since the interpolation of the solution u in the space Xh obviously belongs to thefinite-dimensional subspace Xh, the right-hand side of equation (42) can be boundedin the following way:

infvh∈Xh

∥

∥

∥u− vh

∥

∥

∥

1≤∥

∥

∥u−πh

N (u)∥

∥

∥

1, (43)

where πhN (u) ∈ Xh represents the interpolation of the solution u ∈ X in the space Xh

consisting of polynomials of degree N. The right-hand side of expression (43) canbe further bounded from above if an h− or p−refinement strategy is used to obtaina better approximation of the solution u ∈ X . Since both refinement strategies resultin different estimates for the interpolation error (43) and hence in a different errorestimate (42), they are discussed separately hereafter.

u ∈ Hs (Ω) forsome s≥ 2

∥

∥

∥u−uh

∥

∥

∥

1≤Chl |u|l+1 , (44)

where l = min(N,s−1) and where N represents the approximating order of the C0−spectral elements and where h represents a grid parameter (for example, the maxi-mum of the square root of the area of the spectral elements), which decreases withincreasing number of spectral elements. Here | · |s denotes the semi-norm defined by

|v|2s = ∑|α |=s

‖Dαv‖2L2(Ω) , (45)

where α is a multi-index given by the vector

α = (α1, . . . ,αd) , (46)

where the αi are non-negative integers and

|α|= α1 + · · ·+ αd (47)

We used the following notation for partial derivatives

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Combining expression (43) with the interpolation error corresponding to h−refinement, results in the following estimate if the exact solution


Dα =∂ |α |

∂xα11 ∂xα2

2 . . .∂xαdd

. (48)

For example, if d = 2 and α = (1,2), then

Dα u =∂ 3u

∂x1∂x22

. (49)

The rate of convergence (44) is optimal in the H1−norm since it provides the high-est possible rate of convergence allowed by polynomials of degree N, [73]. Notethat the optimal rate of convergence depends on the polynomial degree (N) and theregularity of the exact solution, denoted by the Sobolev exponent s. A similar rate ofconvergence can be obtained for the L2−norm. In [73], it is shown that the followingL2−error estimate holds if the exact solution u ∈Hs (Ω) for some s≥ 2:

∥

∥

∥u−uh

∥

∥

∥

0≤Chl+1 |u|l+1 . (50)

In case of p−refinement, the order N of the spectral elements is increased whilekeeping the number of the spectral elements constant. Consequently, the grid pa-rameter h remains constant. As a consequence, the convergence rates (44) and (50)are not suitable in this case. In order to obtain useful convergence rates in case ofp−refinement, one can use the Legendre interpolation operator to bound the inter-polation error u−πh

N (u) from above. Assuming that u ∈ Hs (Ω) for some s ≥ 2, itcan be proven [73, page 126] that

∥

∥

∥u−πh

N (u)∥

∥

∥

k≤CNk−s |u|s , with k = 0,1 , (51)

where πhN (u) represents the Legendre interpolation operator applied to the exact

solution u. Combining the latter expression with (42) and (43) results into the fol-lowing error estimate

∥

∥

∥u−uh

∥

∥

∥

k≤CNk−s |u|s , with k = 0,1 . (52)

Note that the rate of convergence is only bounded by the smoothness degree of thesolution s and not by any other grid parameter h as it occurs in the finite elementcase. As a consequence, exponential convergence rates can only be obtained forsmooth problems if a p-refinement strategy is used. Since this rate of convergenceresults from p-refinement, it is called p-convergence hereafter.

Least-squares formulations based on the norm-equivalence (41) are called fullyH1-coercive formulations and they have optimal h−convergence rates in the H1-and L2-norms. Moreover, if the fully H1-coercive least-squares method is supple-mented with strongly imposed boundary conditions, one can use standard finite andspectral elements. Consequently, it is not so surprising that fully H1-coercive for-mulations with strongly imposed boundary conditions are the preferred setting for

189


least-squares finite and spectral element methods since these methods are both prac-tical and optimally accurate.

Note that the above a priori estimates only depend on continuity and coercivity ofthe differential operator L and interpolation estimates. In [69, 70] these error esti-mates have been confirmed numerically with the use of Legendre polynomials. Butsince the estimates are independent of the type of orthogonal basis functions used,similar results will hold for polynomials such as Chebyshev polynomials. These er-ror estimates will play a role in hp-adaptive schemes to be discussed in Section 7.1.

Unfortunately, not all differential equations can be converted into H1-coerciveleast-squares formulations. Especially compressible, inviscid flows which exhibitshocks or contact discontinuities and models like Burgers’ equation do not fit in theH1-coercive framework and therefore require a different treatment to be discussedin Section 6.

5 Incompressible flows

For incompressible flows we can distinguish three regimes, the case where theReynolds number is so low that the convective terms can be neglected (Stokes flow),the case where the Reynolds number is sufficiently small to yield steady solutionsand for slightly higher Reynolds numbers, we have time-dependent flow.

Stokes flow has been thoroughly discussed by Proot, [69, 70]. All function spacesfor which norm-equivalence can be established have been identified in these papers.In this chapter we will show some applications to steady and unsteady incompress-ible, viscous flow problems.

5.1 Governing equations

In terms of the primitive variables (u, p) the governing equations read

∂u∂ t

+(u ·∇)u =−∇p +1

Re∆u+ f in Ω , (53)

∇ ·u = 0 in Ω , (54)

where u represents the velocity vector, p the kinematic pressure, f the forcing termper unit mass (if applicable) and Re the Reynolds number.

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5.2 The first order formulation of the Navier-Stokes equations

In order to obtain an equivalent first order formulation of the unsteady Navier-Stokesequations, the vorticity ω has been introduced as an auxiliary variable. By using theidentity ∇×∇×u =−∆u+ ∇(∇ ·u) and by using the incompressibility constraint∇ ·u = 0, the governing equations subsequently read

∂u∂ t

+ u ·∇u+ ∇p +1

Re∇×ω = f in Ω , (55)

ω−∇×u = 0 in Ω , (56)

∇ ·u = 0 in Ω , (57)

where, in the particular case of the two-dimensional problem, uT = [u1,u2] repre-sents the velocity vector, p the kinematic pressure, fT = [ f1, f2] the forcing term perunit mass (if applicable) and Re is the Reynolds number.

5.3 Linearization of the non-linear terms

Before the least-squares principles can be applied and the corresponding weak formdiscretized with spectral elements, the convective term u ·∇u must be linearized. Tothis end, one can use a Picard (e.g. successive substitution)

u ·∇u≈ u0 ·∇u, (58)

or a Newton linearization

u ·∇u≈ u0 ·∇u+ u ·∇u0−u0 ·∇u0. (59)

In the latter two equations, the subscript “0” indicates that the value of the corre-sponding variable is known from the previous iteration step.

If Picard linearization is used the following linearized momentum equation isobtained:

∂u∂ t

+(u0 ·∇u+ ∇p + ν∇×ω− f) = 0 . (60)

For steady problems the time derivative can be dropped. For time-dependent calcu-lations either a time-stepping scheme needs to be selected or space-time elementscan be used, [19, 20, 21, 26, 63].

The least-squares formulation now becomes: Find u, p,ω ∈ H1(Ω) which mini-mizes the functional in the absence of body forces

I (u, p,ω) = ‖∇ ·u‖2L2 +‖ω−∇×u‖2L2 +

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∥

∥

∥

∥

∂u∂ t

+(u0 ·∇u+ ∇p + ν∇×ω)

∥

∥

∥

∥

2

L2. (61)

Variational analysis with respect to the four unknowns u, p and ω leads to the sym-metric positive definite system.

5.4 Steady flow around a cylinder at low Reynolds

One of the test cases considered for LSQSEM is the steady flow around a circularcylinder. This research was performed by De Groot, [41], with the conventionalleast-squares method discussed above and Direct Minimization, see Section 7.2.

Here we are simulating the flow around a circular cylinder placed perpendicularlyin a uniform parallel flow, see Fig. 1.

Fig. 1 Description of the flow problem

It is known that for flows with Reynolds numbers Re < 45 a steady solution existsand for Reynolds number higher than Re = 45 time-dependent behaviour sets in andthe well-known Von Karman vortex street develops. The investigation of the flowwill be in the Reynolds range: 1 < Re < 45. In this range two different flow typesare observed. For Re < 6 the flow is completely attached to the cylinder as can beseen in Fig. 2 (a). For Re > 6 the flow starts to separate somewhere on the cylinderforming two attached eddies behind the cylinder, Fig. 2 (b). For a benchmark be-tween LSQSEM on the one hand and experimental data on the other, the estimationof the Re number for which the twin vortices appear, is an important property of theflow. This Re number is from here on referred to as Reonset . The twin vortices growin length and change shape as Re increases. The accurate prediction of this relationis a common benchmark. Fig. 3 gives a schematic overview of the flow where twosteady vortices exist behind the cylinder.

192


(a) (b)

Fig. 2 Types of flow for low Re numbers, Re < 6 (a) and Re > 6 (b)

Fig. 3 Flow characteristics for a low Re flow around a circular cylinder

5.5 Comparison of numerical data with experimental data

Results of Taneda [81], Coutanceau [17] and Tritton [82] are used to benchmark theresults of LSQSEM.

5.5.1 S/d: LSQSEM vs. Experiment

The LSQSEM values are compared to experimental values from Coutanceau andTaneda in Table 1. The same data are depicted in Fig. 4. For the LSQSEM simula-tions a mesh consisting of 62 elements was used with varying polynomial degree.The LSQSEM results given in Fig. 4 were obtained with polynomial degree P = 10in both x- and y-direction. This corresponds to an 11th-order scheme.

The Onset Reynolds number The onset Reynolds number is the lowest Reynoldsnumber for which the twin vortices behind the cylinder can be observed. By usingpolynomials of degree N = 10, Reonset is sought on the mesh where symmetry isexplicitly imposed. Fig. 5 displays S/d for various Reynolds numbers.

Vortex center The vortex center, of which the location is indicated by a and bin Fig 3, is also a property commonly investigated for this type of flow. The resultfor a and b can be found in Table 3 for the simulations as well as the experimentalvalues of Coutanceau [17].

193


Table 1 Recirculation length S/d for LSQSEM and data from Coutanceau

Re S/dGroot Coutanceau Taneda

10 0.27 0.34 0.320 0.94 0.93 0.930 1.57 1.53 1.540 2.17 2.13 2.1

Re [-]

S/d

[-]

5 10 15 20 25 30 35 40 45

0.5

1

1.5

2

Groot, P10CoutanceauTaneda

Fig. 4 Recirculation length S/d form LSQSEM and data form Coutanceau

Table 2 Recirculation length S/d, obtained by the least squares method and other numerical meth-ods

Re S/dGroot Groot Den & Shi Kaw & Jai Tak & Kel Nie & Kel

2003 1965 1966 1969 1973

10 0.27 0.25 0.56 0.3 0.25 0.21720 0.94 0.88 1.06 1 0.935 0.80330 1.57 1.44 1.16 1.75 1.611 1.54340 2.17 1.94 0.94 2.515 2.325 2.179

194


Re

S/d

5 5.5 6 6.5 70

0.01

0.02

0.03

0.04

0.05

0.06

0.07

S/d

Fig. 5 Recirculation length in the vicinity of Reonset

Table 3 Position of vortex core as a function of Re

Re Groot, P = 10 Coutanceaua b a b

10 0.12 0.24 0.12 0.3220 0.35 0.44 0.33 0.4730 0.53 0.52 0.55 0.5440 0.68 0.58 0.76 0.59

Separation angle The separation angle is also a characteristic of the flow arounda circular cylinder. For the definition of the separation angle see Fig. 3. Table 4shows the results for the separation angle in the range, 10≤ Re≤ 40, for two differ-ent settings. Table 4 shows in the third column the experimentally obtained valuesof Coutanceau [17].

Table 4 Separation angle as function of Re

Re Groot CoutanceauP = 7 P = 10

10 29.75 30.14 32.520 43.50 43.93 44.830 49.57 49.90 50.140 53.38 53.77 53.5

195


Drag coefficient Besides the flow field features as separation angle and recircu-lation length, also an integral quantity of the flow is investigated, the drag coefficient(CD).

Almost every textbook on the fundamentals of Aerodynamics gives ”accurate”formulae to predict CD for a cylinder in uniform parallel flow. Let us investigate howwell the least squares solution matches these curve fitted formulae. The first formulais derived by Tritton [82] and valid around Re≈ 1.0,

CD ≈ 1 +10.0

Re2/3D

. (62)

The second formula, from Sucker and Brauer [80], is valid over a much wider range,10−4

< Re < 2.0 ·105,

CD ≈ 1.18 +6.8

Re0.89D

+1.96

Re1d/2−

0.0004 ·ReD

1 + 3.64 · e−7 ·Re2D

. (63)

White [83] emphasizes the good accuracy of (63). Fig. 6 shows the converged resultsfrom the simulation together with the curve fits. Despite the claimed accuracy of thecurve fits a significant gap with the numerical data emerges. To further investigatethe accuracy of the numerically obtained CD values, experimental data of Tritton[82] was used. Fig. 7 shows the numerical results obtained with the least squaresmethod and the experimental results from Tritton [82] over the range: 1≤ Re≤ 45.

Re

Cd

[-]

0 10 20 30 40 501

1.5

2

2.5

3

3.5

4

4.5

5

Sucker BrauerTrittonGroot, P=7

Fig. 6 Numerical results for CD and curve fits form experimental data

196


Re

Cd

[-]

0 10 20 30 400

2

4

6

8

10

12

14

Groot, P7Tritton

Fig. 7 Drag coefficient (CD) from simulation and experiments

5.6 Unsteady flow around a cylinder at low Reynolds

For time dependent flows one can either employ time stepping methods or discretizethe system of governing equations in space-time. The latter approach was investi-gated by De Maerschalck, [18, 19, 20, 21]. Kwakkel, [49, 53] performed a series ofnumerical simulations for the flow around a cylinder in a channel. A BDF3 schemewas used for the time integration.

The final test case is the periodic flow around a moving circular cylinder in anarrow channel. To be able to simulate the flow around the cylinder, the cylinder isfixed and the channel is moved (change of reference frame). The boundary condi-tions are u = 1 and v = 0 at the inflow and channel walls and a no-slip condition(u = v = 0) at the cylinder surface. The outflow boundary condition is the same asin the work of Pontaza, [58] and is described in Section 5.6.1.

5.6.1 Outflow boundary condition

The outflow boundary conditions are defined as(

−p +1

Re∂u∂x

)

nx +1

Re∂u∂y

ny = 0, (64)

1Re

∂v∂x

nx +

(

−p +1

Re∂v∂y

)

ny = 0, (65)

197


where nx and ny are the x and y-components of the outward unit normal along theboundary. This boundary condition is enforced in a weak sense through the least-squares functional, see (32).

Vortex shedding is described by the dimensionless Strouhal number

St =f lV

, (66)

where f is the frequency of the vortex shedding, l the characteristic length (diameterof the cylinder) and V the free stream velocity of the fluid. Pontaza, [58], reports aStrouhal number of St=1/1.88=0.5319, see Fig. 8. The Reynolds number was set toRe = 100.

(a) Pressure contours.

(b) Vorticity contours.

Fig. 8 Instantaneous pressure and vorticity contours for Px = 8.

5.6.2 Base pressure coefficient

The base pressure coefficient Cpb is defined by

Cpb =pb− p∞

q∞, (67)

where pb is the base pressure, p∞ the free stream pressure and where

q∞ =12

ρ∞V 2∞

is the dynamic pressure.

198


0 1 2 3 4 5 6 7 8 9 10−13.0

−12.8

−12.6

−12.4

−12.2

−12.0

Time

Cpb

Fig. 9 Time history of Cpb P = 6 (red) P = 8 (blue) and the reference value from [58] with P = 6(green)

Fig. 9 shows the time history of the base pressure coefficient Cpb for two poly-nomial degrees and the reference solution from [58].

5.6.3 Drag and lift coefficients

The lift CL and drag CD coefficients are often used to compare the results of the flowaround a circular cylinder. These coefficients are defined by

CD =Fx

dq∞, CL =

Fy

dq∞, (68)

where F is the force in the direction indicated, q∞ the dynamic pressure and d thediameter of the cylinder, which is equal to unity. The forces are calculated by

F =−∫

ΓpdΓ +

∫

Γτ ·dΓ , (69)

where Γ is the boundary of the cylinder and τ the extra stress tensor. In componentsthis can be written as

Fx =−∫

ΓpnxdΓ +

∫

Γ(τxxnx + τxyny)dΓ , (70)

Fy =−∫

ΓpnydΓ +

∫

Γ(τyxnx + τyyny)dΓ . (71)

The extra stress tensor for a Newtonian incompressible fluid is

τ = µ

[

2 ∂u∂x

∂u∂y + ∂v

∂x

∂v∂x + ∂u

∂y 2 ∂v∂y

]

, (72)

where µ is the dynamic viscosity. For Re < 188 the flow will be periodic in time,so both CD and CL must be periodic. The frequencies of the drag and lift curvesboth depend on the frequency of the vortex shedding. The frequency of the drag

199


curve has twice the frequency of the lift curve however. This is because for the liftit is important which vortex is shed off (top or bottom) and for the drag it is onlyimportant if a vortex is shed. Comparison of Fig. 10 and 11 shows that this is thecase.

0 1 2 3 4 5 6 7 8 9 1012.2

12.3

12.4

12.5

12.6

12.7

Time

CD

Fig. 10 Time history of CD; P = 6 (red) P = 8 (blue) and the reference value from [58] with P = 6(green)

0 1 2 3 4 5 6 7 8 9 10−3.0

−2.0

−1.0

0.0

1.0

2.0

3.0

Time

CL

Fig. 11 Time history of CL; P = 6 (red) P = 8 (blue) and the reference value from [58] with P = 6(green)

Fig. 11 shows that the results for CL agree with the reference values. The fre-quency that follows from (66) is St = 1/1.88 = 0.5319, which agrees with the valueof [58]. The value of 1.88 is obtained from the time history plot of the lift coefficientand is the time between two peaks, see Fig. 11.

6 Compressible flows

Inviscid, compressible flows may exhibit discontinuous solutions – shocks or con-tact discontinuities. These solutions are not in H1(Ω) and therefore an H1-coerciveformulation is not possible. The proper functional setting for these problems is in theweaker space H(div;Ω). Since in the finite-dimensional case all norms are equiva-lent, one can still use the conventional least-squares approach but in this case opti-mal convergence is not guaranteed.

200


Compressible flows in the absence of dissipative terms are governed by the Eulerequations. There are several ways in which the Euler equations in differential formcan be written, but only the conservative form in terms of conserved quantities willbe presented.

The two-dimensional Euler equations in conservation form are given by

∂∂ t

ρρuρvρE

+∂∂x

ρuρu2 + p

ρuvρuH

+∂∂y

ρvρuv

ρv2 + pρvH

=

0000

. (73)

These equations express conservation of mass, conservation of momentum in the x-and y-direction and conservation of energy, respectively. Here ρ is the local density,p is the pressure and (u,v) denotes the fluid velocity. The total energy per unit massis denoted by E . The total energy can be decomposed into internal energy e and thekinetic energy per unit mass

ρE = ρe +ρ2

(

u2 + v2)=p

γ−1+

ρ2

(

u2 + v2), (74)

where in the last equality we assume a calorically ideal, perfect gas with γ the spe-cific ratio of heats of the gas.

6.1 Compressible flow over a circular bump

In this section results are given for the flow over a circular bump in a 2D channel.Results will be given for subsonic flow, M∞ = 0.5, transonic flow, M∞ = 0.85 andsupersonic flow, M∞ = 1.4. This is a difficult test problem over the entire Machrange for spectral methods due to the presence of stagnation points at the leadingand trailing edge of the bump. See [38] for further details of this approach.

6.1.1 General geometry and boundary conditions

The general geometry for the channel flow with a circular bump is shown in Fig. 12.The bump is modeled by curved elements using the transfinite mapping by Gordonand Hall, [40]. All length and height parameters of the channel will be scaled withthe chord length c of the bump.

The influence of the mesh is assessed by refining the mesh around the stagnationpoint. The refined mesh consists of 72 elements, Fig. 13.

The entropy variation s in the domain is calculated with the freestream entropyas a reference:

s =s− s∞

s∞, where s = pρ−γ

. (75)

201


c

h

Fig. 12 The general geometry of the 2D channel with a circular bump

(a) Mesh

(b) Details of mesh near the bump

Fig. 13 Refined mesh near stagnation points consisting of 72 spectral elements of polynomialdegree N = 4. (a) Spectral element mesh, (b) close-up near the bump:spectral elements with Gauss-Lobatto-grid

Pressure contours are given in Fig. 14.

0.99

0.94

0.92

1.02

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Fig. 14 Pressure contours for the subsonic flow

202


The results along the lower wall for a polynomial degree N = 4, integration orderP = 5, are displayed in Fig. 15. This figure shows that the resolution of the stagnationpoints is very pronounced and the flow almost retains its inflow Mach number afterthe bump. The entropy change remains very small over the bump and the artificialentropy increase is restricted to the location of the stagnation points.

X

M

0 0.5 1 1.5 2 2.5 30.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) Mach number

X

Ent

ropy

0 0.5 1 1.5 2 2.5 3

0

0.005

0.01

0.015

0.02

(b) Entropy

Fig. 15 The Mach number and entropy distribution for the subsonic flow on a refined mesh

6.1.2 Results for transonic flow

To investigate the transonic flow over a bump the geometry is the same as that forthe transonic flow problems described by Spekreijse, [78], and Rizzi and Viviand,[76].

As in the subsonic case, the chord length of the bump is c = 1. The length of thechannel however, is 5 times the chord length and the height is set at 2.073 times thechord length. The height of the bump is 4.2% of the chord length. The mesh usedfor this test case is shown in Fig. 16. The polynomial degree is N = 5 whereas theintegration order is P = 6.

In Fig. 17 the Mach contours at an inflow Mach number of M = 0.85 are com-pared to the finite volume results produced by Spekreijse, [78].

The shock is positioned at approximately 86% of the bump with the Mach num-ber just upstream of the shock being M ≈ 1.32. These results are quantitatively inagreement with the finite volume results obtained by Spekreijse. In Fig. 18 the Machnumber distribution along the lower wall of the channel is shown.

203


0 1 2 3 4 5

0

0.5

1

1.5

2

Fig. 16 The mesh used for the transonic test case. The height of the bump is 4.2% of the chordlength and 100 elements are used.

x

y

0 1 2 3 4 50

0.5

1

1.5

2

x

y

0 1 2 3 4 50

0.5

1

1.5

2

Fig. 17 Iso-Mach lines for transonic flow, M = 0.85, obtained by LSQSEM (green) and FiniteVolume Method by Spekreijse [78], (black)

x

M

0 1 2 3 4 50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Fig. 18 Mach number distribution along the lower wall of the channel for a M = 0.85 flow:LSQSEM solution (green line) and Finite Volume results obtained by Spekreijse, [78], (black line)

204


6.1.3 Results for supersonic flow

The geometry used for the supersonic test case is similar to that considered for thesubsonic flow test case. The only difference is the height of the bump which is 4%of the chord length for this test case. The mesh has a total of 120 elements as can beseen in Fig. 19.

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

Fig. 19 The mesh used for the supersonic test case. The height of the bump is 4% of the chordlength and 120 elements are used.

At inflow the density is set to ρ = 1.4 and the pressure to p = 1. The Machnumber at the inlet boundary is M = 1.4.

At the leading edge of the bump a shock develops and runs into the domain untilit is reflected by the upper wall. From the trailing edge a shock originates at a slightlysmaller angle than the shock at the leading edge. In the region behind the bump thetwo shocks collide and then merge into a single shock. The iso-Mach contours forthis test case are shown in Fig. 20 together with the results obtained by Spekreijse,[78]. This figure reveals that the shock structures agree.

x

y

0 1 2 30

0.2

0.4

0.6

0.8

1

x

y

0 1 2 30

0.2

0.4

0.6

0.8

1

Fig. 20 Iso-Mach lines and shock structure obtained by LSQSEM (green) and Finite VolumeMethod by Spekreijse [78], (black)

205


7 Miscellaneous topics

In this section several topics will be discussed which improve the performance ofLSQSEM.

7.1 hp-adaptive LSQSEM

Although the combination of the least-squares formulation and the spectral ele-ment technique leads to a very robust scheme which does not require any formof stabilization and renders highly accurate solutions, using high order polynomi-als throughout the entire computational domain is very expensive. The followingprocedure is described in [33].

7.1.1 The mortar element method

Mortar

Computational domain

Ω1

Ω2 Ω3

Ω2 Ω3

Ω1

γ

Fig. 21 The mortar element approach - patching the element edges with oneapproximation space

In the mortar element method (MEM), neighbouring elements in Rd are patchedtogether by mortar-like elements in Rd−1. In R2 the mortar elements consist of linesegments as sketched in Fig. 21. The ith boundary of element k, denoted by Γ k

i , isassociated with a number of mortars, γ j. The solution on the mortars, φ , is connectedto the solution at the border of the two neighboring elements. This establishes aconnection between the solution at the edge of an element, denoted by ub, and themortar solution φ . If we choose a polynomial approximation for φ on the mortar, wecan express the expansion coefficients at the boundary of the element, ub, in termsof the expansion coefficients of the solution on the mortar, φ , as

ub = Zφ . (76)

206


The precise relation is inconsequential as long as the matrix Z is of full rank forat least one of the elements associated with the mortar. This condition prevents theappearance of spurious mortar solutions.

Having established the relation between the elemental boundary unknowns andthe mortar unknowns, we can express the global system in terms of the inner elementunknowns, ui and the mortar unknowns φ only:

uk =

[

ub

ui

]

=

(

Z 00 I

) [

φui

]

=[

Zk]

uk, (77)

where uk represents the true unknowns, i.e. the projected mortar values and theinternal unknowns.

This transformation converts the least-squares formulation into

LTWLuk = LTW fk ⇐⇒[

Zk]T

LTWL[

Zk]

uk =[

Zk]T

LTW fk. (78)

Assembling all element contributions by summing over the projected element ma-trices gives the global system to be solved.

In this paper the solution on the mortar is defined by an L2-projection of theelement boundary solution

∫

Γ kl

(u|Ω k −φ) ψ ds = 0, ∀ sides l and k = 1, . . . ,K , (79)

where ψ ∈PM(

Γ kl

)

and M is the polynomial degree of the mortar solution. M shouldbe greater than or equal to the degree of the solution in the adjoining elements toprevent spurious mortar solutions. When the Lagrangian basis functions, defined inSection 3, are employed, the vertex condition (Maday et al. [56]) is automaticallysatisfied. For a more extensive treatment of the mortar element method the reader isreferred to [48, 56].

7.1.2 The error estimator

Having discussed how to match spectral elements with different size and polyno-mial representation, we now have to find a way to detect those elements that needrefinement.

In the least-squares formulation we select the solution which minimizes the resid-ual globally over the whole domain Ω . Having obtained such a minimizing solutionwe can evaluate the least-squares functional locally over every sub-domain A⊂ Ω .This gives

η2A = ‖L uh− f‖2

L2(A) = ‖L(

uh−uex

)

‖2L2(A) , (80)

due to the linearity of L . Well-posedness of the problem (30) then implies that

α2‖e‖2X(A) ≤ η2

A ≤C2‖e‖2X(A) , (81)

207


where e = uh−uex. This means that the effectivity index θA,X , defined by

θA,X =ηA

‖e‖X(A), (82)

which compares the estimated error ηA to the exact error in the X -norm is boundedby

α ≤ θA ≤C . (83)

Alternatively, we may compare the estimated norm ηA with the residual norm‖R‖L2(A) using the fact that the residual norm is norm equivalent to ‖e‖X . Denot-ing this effectivity index by θA,R gives the rather trivial result

θA,R :=ηA

‖L uh− f‖Y(A)≡ 1 . (84)

Based on this observation ηA will be used to identify those regions (elements incase A = Ω k) which are selected for refinement. This estimator has also been usedby Liu, [54] and Berndt et al., [4].

7.1.3 Estimation of the Sobolev regularity

Having found a way to match functionally and geometrically non-conforming ele-ments and an indicator ηΩ k which flags elements for refinement, we now have todetermine how to refine. This choice is based on the smoothness of the underlyingexact solution. If the exact solution is locally sufficiently smooth, polynomial en-richment is employed. However, if on the other hand, the underlying exact solutionhas limited smoothness h-refinement is used.

Let κ be a spectral element with size parameter hκ and polynomial degree pκ .Let uκ

ex be the exact solution in that element, where uκex ∈Hkκ , where kκ ≥ 0 denotes

the Sobolev regularity of the exact solution. Let uhκpκ denote the LSQSEM solution

with uhκpκ ∈ Hq, 0≤ q≤ kκ then

‖uκex−uhκ

pκ‖Hq ≤C(hκ)sk−q

(pκ)kκ−q ‖uκex‖Hkκ , (85)

where sκ = min(pκ + 1,kκ) and C is a generic constant. This error estimate tellsus that if the solution is very smooth (kκ very large) then the error decreases morerapidly by increasing pκ in the denominator. For practical purposes the functionis considered smooth if kκ > pκ + 1 and non-smooth when kκ ≤ pκ + 1, in whichh-refinement is more effective.

So the choice between h-refinement and p-enrichment is dictated by the Sobolevindex of the exact solution. Although the exact solution is in general not available,we can still estimate this index from its numerical approximation. Houston et al.,[50] have developed a method to estimate the Sobolev index from a truncated Leg-

208


endre series. They assume that the one-dimensional solution is in L2(−1,1) whichallows for a Legendre expansion

u(x) =∞

∑i=0

aiLi(x) , with ai =2i+ 1

2

∫ 1

−1u(x)Li(x)dx . (86)

By Parseval’s identity the fact that u∈ L2(−1,1) is equivalent to convergence of theseries

∞

∑i=0

|ai|2 2

2i+ 1. (87)

In [50] it is shown that if

∞

∑i=[k+1]

|ai|2 2

2i+ 1Γ (i+ k + 1)

Γ (i− k + 1), (88)

converges, then u ∈Hkw(−1,1), where

Hkw =

u ∈ L2(−1,1)∣

∣

k

∑j=0

∫ 1

−1

∣

∣

∣D( j)u(x)

∣

∣

∣

2(

1− x2) jdx < ∞

, (89)

for integer values of k. By using the Γ -function in (88) this identity can be extendedto fractional Sobolev spaces, see [50] for details.

Given the Legendre coefficients ai, convergence of the series in (88) can be es-tablished by well-known techniques such as the ratio test, or the root test.

In this work the root test is employed. This leads to the calculation of

lk =log(

2k+12|ak|

2

)

2logk. (90)

If l = limk→∞ lk > 1/2 then u ∈Hl−1/2−εw (−1,1), ∀ε > 0. Else u ∈ L2(−1,1). Since

in a numerical solution only a finite number of Legendre coefficients ai are available,this test is applied to the highest Legendre coefficient available in the numericalapproximation. Based on the estimated Sobolev index l−1/2, the decision is madewhether to refine the mesh, or to increase the polynomial degree locally.

This one-dimensional estimate of the Sobolev index can be extended to multi-dimensional problems by treating each co-ordinate direction separately, see [50] fordetails. Several tests have been performed to establish the validity of this estimator.

Proposition 1. If an element κ at refinement level r with characteristic mesh sizehr

κ and polynomial degree prκ is flagged for refinement by the error indicator, we

calculate lpκ by (90). Then for the Sobolev index kpκ = lpκ −1/2 we have

If kpκ > prκ + 1 then pr+1

κ ← prκ + 1 .

If kpκ ≤ prκ + 1 then hr+1

κ ← hrκ/2 .

(91)

209


7.1.4 Application to the space-time linear advection equation

This section uses the one-dimensional, linear advection problem to validate the pre-sented hp-adaptive theory. The application of LSQSEM to hyperbolic equations hasbeen studied by De Maerschalck, [18, 19, 20, 21].

The model problem is defined as

∂u∂ t

+ a∂u∂x

= 0 with 0≤ x≤ L, t ≥ 0, a ∈ R (92)

u(0,t) = 0 , (93)

u(x,0) =

12 −

12 cos

(

2π x−x0L0

)

if x0 ≤ x≤ x0 + L0

0 elsewhere .

(94)

where a is the advection speed and L is the length of the domain in spatial direction.On the left boundary of the domain a Dirichlet boundary condition, u(0,t) = 0, isimposed. The initial disturbance, u(x,0), is a cosine-hill with offset x0 and width L0.We use a space-time formulation that treats the one-dimensional advection problem

0 0.5 1X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

Fig. 22 Illustration of the unstructured mesh and continuous propagation ofthe cosine-hill with hp-refinement

with a two-dimensional least-squares formulation and considers the time variable t

210


as second spatial variable. Instead of calculating the solution over the whole domainat once, we use a semi-implicit approach. The domain is subdivided into severalspace-time strips for which the solution is approximated using the LSQSEM, see[20] for details. In the following, 32 time strips are used within the domain Ω =[0,1], where each time strip has initially 32 cells. Each time strip uses the solution atthe previous strip as initial condition, except for the first strip that uses the prescribedinitial condition (94). All time strips use the Dirichlet condition (93) on the leftboundary. The advection speed a = 0.85, x0 = 0.13 and L0 = x0 + 0.5. The exactsolution of this problem uex ∈ H5/2−ε(Ω), for all ε > 0. The regularity of the exactsolution is limited in space-time over the lines x−at = x0 and x−at = x0 +L0. Forall other points in the space-time domain the solution is infinitely smooth.

7.1.5 Illustration of an hp-adaptive strategy

Note that even though only the second derivative is discontinuous, the regularityestimator accurately identifies the region with limited regularity as depicted in Fig-ure 22. No h-refinement is used in the smooth parts of the domain.

0 0.5 1X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

8765432

Fig. 23 Polynomial degree distribution for hp-adaptivity within each element

Figure 23 shows the final polynomial degree distribution. We imposed the so-called 1-level adaptivity, where the difference in refinement level between neigh-bouring elements may not be more than one. In the regions where the exact solution

211


is zero neither p-enrichment nor h-refinement is used. Along the cosine hill polyno-mials of degree N = 4 are used, the green strip. The only place where the algorithmuses higher order elements is along the lines with limited regularity. Along theselines both p-enrichment and h-refinement are employed.

The reason high order elements are used along these lines is that in order to pre-dict the Sobolev regularity accurately enough, one needs sufficiently high Legendrecoefficients.

7.2 Direct Minimization

In this section an alternative method will be described which minimizes the residualdirectly in contrast to the conventional least-squares formulation where one employsvariational analysis to set up the weak formulation. The resulting condition numberis only the square root of the condition number that would be obtained if the conven-tional least squares method had been used. In addition, the new method circumventsa costly matrix-matrix multiplication thus avoiding loss of precision and fill-in inthe stiffness matrix. This approach is also described in [49].

7.2.1 Conventional least-squares finite element method

In Section 2 and 3 the approach for the conventional or variational least-squaresformulation is described. This approach can be summarized as

(L (u),L (v))= ( f ,L (v)) ⇐⇒∫

ΩL (u)L (v)dΩ =

∫

ΩfL (v)dΩ ∀v∈X(Ω) .

(95)The next step consists of domain decomposition where the integration over Ω iswritten as the sum of the integrals over the sub-domains Ω k, k = 1, . . . ,K,

∑k

∫

Ω kL (u)L (v)dΩ k = ∑

k

∫

Ω kfL (v)dΩ k ∀v ∈ X(Ω k) . (96)

∑k

[

∑i

uN,ki

∫

Ω kL (φi)L (φ j)dΩ k

]

= ∑k

∫

Ω kfL (φ j)dΩ k ∀φ j , j = 1, . . . ,N .

(97)It suffices in (97) to take v = φ j, because L is assumed to be a linear operator andsince any arbitrary v in the finite-dimensional subspace is a linear combination ofthese basis functions.

Inserting the Gauss-Lobatto integration then gives

212

N,k=∑iuN,ki φi(x),

where the φi are basis functions, which span the finite-dimensional subspace overΩ k

Then we insert the finite-dimensional approximation for each element u


∑k

[

∑i

uN,ki ∑

pL (φi)(xp)L (φ j)(xp)wp

]

= ∑k

∑p

f (xp)L (φ j)(xp)wp ,

∀φ j , j = 1, . . . ,N . (98)

Here xp denote the Gauss-Lobatto points and wp the Gauss-Lobatto weights definedby (36) and (40), respectively. Note that in the multi-dimensional case xp is a vector,φi is a tensor product and wp is the product of the Gauss-Lobatto weights in eachdirection separately. We now define in each element the matrix Ak by

(

Ak)

pi= L (φi)(xp) , (99)

i.e. the matrix coefficient denotes the application of the differential operator to theith basis function evaluated at the pth Gauss-Lobatto point. Furthermore we introducethe diagonal weight matrix W k by

(

W k)

pp= wp . (100)

The discretized least-squares problem (98) can then be written as

∑k

[

(

Ak)T

WAk]

uN,k = ∑k

[

(

Ak)T

W F

]

, (101)

where the vector F contains the elements (F)p = f (xp). The system of algebraicequations obtained in this way, i.e. using variational analysis, is called the normalequations. The normal equations reflect on a discrete level the symmetry that wasalready mentioned at the continuous level. Note that Gauss-Lobatto integration maybe performed on a finer grid than the grid on which the unknowns are defined. Inthis case the matrix Ak is non-square, i.e. there are more rows than columns in thematrix. The resulting normal equations, however, deliver a square, positive definitematrix which possesses a unique solution.

7.2.2 Direct Minimization - LSQSEM-DM

In order to avoid variational analysis we start with the original minimization prob-lem.

Find u ∈ X(Ω) which minimizes I (u) =12‖L u− f‖2

Y(Ω) . (102)

Since we decompose the computational domain Ω into a union of non-overlappingsub-domains Ω k, k = 1, . . . ,K, we can also write this as

Find all uk ∈ X(Ω k) which minimize the functional

213


I (u1, . . . ,uK) =

K

∑k=1

∥

∥

∥L uk− f

∥

∥

∥

2

Y (Ω k). (103)

Now in each domain Ω k we are going to restrict our search to a finite-dimensionalsubspace XN(Ω k)⊂ X(Ω k) using the spectral approximation given by (38)

Find all uN,k ∈ XN(Ω k) which minimize the functional

I (uN,1, . . . ,uN,K) =

K

∑k=1

∥

∥

∥L uN,k− f

∥

∥

∥

2

Y (Ω k). (104)

Next we introduce numerical quadrature to evaluate the integrals which constitutethe L2-norm. This gives

Find all uN,k ∈ XN(Ω k) which minimize the functional

I (uN,1, . . . ,uN,K)≈

Nkint

∑p=0

K

∑k=1

(

L uN,k− f)2∣

∣

∣

∣

xp

wp , (105)

where Nkint denotes the number of integration points in element k. Introducing our

matrix notation (99) and (100) this can be written as

Find all uN,k ∈ Xh(Ω k) which minimize the functional

K

∑k=1

(

AkuN,k−Fk)T

W k(

AkuN,k−Fk)

=K

∑k=1

‖√

W k(

AkuN,k−Fk)

‖2. (106)

So the procedure of domain decomposition, insertion of an approximate solutionand the use of numerical integration has converted the minimization in the func-tion space L2(Ω) to a minimization problem in Euclidean space: Find the finite-dimensional vector fields u = (u1

, . . . ,uK)T ∈ Rn such that the norm in Rm givenby (106) is minimized. If m = n, i.e. the number of unknowns in the global systemequals the number of equations, the use of the weight matrix W k is then inconse-quential and the problem reduces to a collocation method evaluated in the Gauss-Lobatto-Legendre points, [42, 43, 44], given by

K

∑k=1

(

AkuN,k−Fk)

= 0 . (107)

In case m > n, we have more equations than unknowns and the solution whichminimizes the residual norm of the overdetermined system is then given by

K

∑k=1

√

W kAkuk =K

∑k=1

√

W kFk. (108)

214


Let us for convenience introduce the following notation B = ∑Kk=1

√W kAk and G =

∑Kk=1

√W kFk. Then we have the following Theorem, [5, 6]:

Theorem 2. Let B ∈ Rm,n and G ∈ Rm, then the following 2 statements are equiva-

lent:

• Determine the vector u ∈ Rn which minimizes the Euclidean norm ‖Bu−G‖2;• Determine the vector u ∈ Rn such that the residual R = G−Bu ∈N

(

BT)

.⊓⊔

See [49] for the proof.The above Theorem shows that finding the minimizer of the overdetermined sys-

tem (108) is equal to imposing

(

K

∑k=1

√

W kAk

)T(√

W k(

Akuk−Fk))

= 0

⇐⇒ (109)

K

∑k=1

(

Ak)T

W k(

Ak)

u =K

∑k=1

(

Ak)T

W kFk,

which is the same equation that we obtained using variational analysis. Therefore,direct minimization given by (108) is equivalent to (101) as a result of the Theorem.

However note that (108) is more appealing to use than (101). Since no pre-multiplication is employed we do not lose the sparsity pattern of the matrix Ak andwe prevent fill-in in the global matrix. Bear in mind that W k is a diagonal matrixand so is its square root. Pre-multiplication with a diagonal matrix amounts to row-scaling which does not affect the sparsity.

7.2.3 Global QR

Let us return to our global system of algebraic equation given by

Bu = G ⇐⇒ Find u which minimizes ‖Bu−G‖2, (110)

where B ∈ Rm,n, u ∈ Rn and G ∈ Rm. Now for any orthogonal matrix Q ∈ Rm,m wehave

‖Q(Bu−G)‖2 = ‖Bu−G‖2, (111)

since the Euclidean norm is invariant under orthogonal transformations.We now decompose the m×n matrix B in a QR-decomposition, B = QR, where

Q is an orthogonal m×m-matrix and R is an m× n upper-triangular matrix. The Rmatrix can be written as

R =

(

R0

)

, (112)

215


where R is an upper-triangular n×n matrix with non-zero diagonal entries, and 0 isan (m−n)×n matrix with zero entries.

With this decomposition we have

‖Bu−G‖2 = ‖QT (Bu−G)‖2

= ‖Ru−QT G‖2

=

∥

∥

∥

∥

(

R0

)

u−

(

c1

c2

)∥

∥

∥

∥

2

= ‖Ru− c1‖2 +‖c2‖

2, (113)

where c1 is an n-vector and c2 is an m−n-vector. With this decomposition, minimiz-ing the Euclidean norm is straightforward. The second term, ‖c2‖

2, in (113) cannotbe minimized. The only terms that can be made small – zero in fact – is the firstterm on the right hand side of (113). So we have for the least-squares solution

uLS = R−1c1 , (114)

which is just a back-substitution for the upper-triangular matrix R. An approxima-tion to the L2-norm of the residual is given by the second term, ‖c2‖

2, and thisvalue is available without solving for uLS. This may be advantageous in hp-adaptiveschemes.

Note again, that when exact arithmetic is used the minimizer uLS is equal to theleast-squares solution obtained by the conventional least-squares formulation whichis obtained by applying variational analysis and solving the normal equations. Thisalgorithm can be improved when one observes that it is not necessary to computethe matrix Q to solve the over-determined system directly.

With a suitable global node numbering this algorithm can be converted into ablock-QR algorithm. See [49] for further particulars.

7.2.4 The Poisson equation

In this section a sample problem is presented which consists of a modified Poissonequation given by

κ∆φ = f (x,y) , (x,y) ∈ [−1,1]2 , (115)

wheref (x,y) =−2κ sinxsin y . (116)

The solution in this case is obviously independent of the parameter κ , but the con-dition number of the resulting system will strongly depend on κ .

In order to apply the least-squares formulation which allows for a C0 finite ele-ment approximation, the governing equation needs to be rewritten as an equivalentfirst order system

u−∇φ = 0 , (117)

216


κ∇ ·u = f . (118)

Note that there are other equivalent first order systems possible with improved sta-bility estimates, but this model problem is only introduced to compare formulations.

Following Jiang, [51], it is easy to show that this problem is well-posed

κ2C(

‖φ‖2H1(Ω) +‖u‖2

H(div;Ω)

)

≤ ‖u−∇φ‖2L2(Ω) +‖κ∇ ·u‖L2(Ω)

≤ ‖φ‖2H1(Ω) +‖u‖

2H(div;Ω)

, (119)

for κ ≤ 1. So the coercivity constant scales with κ2 and therefore the conditionnumber is proportional to κ−2. We therefore expect to see differences between theconventional least-squares formulation and direct minimization as proposed in thissection for κ ≪ 1. For κ = O(1), however, both formulations are expected to givesimilar results. In order to assess the improved stability of direct minimization theartificially ill-conditioned system is solved on a 5× 5 grid with polynomial degreeN = 5.

-1

-0.5

0

0.5

1

phi

-3

-2

-1

0

1

2

3

X

-3

-2

-1

0

1

2

3

Y

X Y

Z

-1

-0.5

0

0.5

1

phi

-3

-2

-1

0

1

2

3

X

-3

-2

-1

0

1

2

3

Y

X Y

Z

Fig. 24 Solution for κ = 1 obtained by the conventional least-squares formulation (left) and theresult obtained by direct minimization (right)

Figure 24 (left) shows a plot of the solution of the Poisson equation obtained bythe conventional least-squares formulation with κ = 1. Figure 24 (right) gives thesolution obtained by Direct Minimization for κ = 1. The results are indistinguish-able. This follows from the observation that both methods are equivalent if exactarithmetic is used.

In Figure 25 results for the case κ = 10−5 are presented, where the differencesbetween the conventional least-squares formulation and direct minimization becomeapparent. The conventional least-squares formulation is unable to approximate theexact solution sufficiently accurate due to the loss of precision, whereas Direct Min-imization still approximates the solution sufficiently accurate.

The L2-error for both the conventional least-squares formulation and Direct Min-imization versus the parameter κ is depicted in Figure 26. The conventional least-squares formulation (red line) approximates the solution rather well for a κ up to10−4 after which the error grows dramatically. Direct Minimization is capable of

217


-1

-0.5

0

0.5

1

ph

i

-3

-2

-1

0

1

2

3

X

-3

-2

-1

0

1

2

3

Y

X Y

Z

-1

-0.5

0

0.5

1

ph

i

-3

-2

-1

0

1

2

3

X

-3

-2

-1

0

1

2

3

Y

X Y

Z

Fig. 25 Solution for κ = 10−5 obtained by the conventional least-squares formulation (left) andthe result obtained by direct minimization (right)

10-1410-1210-1010-810-610-410-2100

k

10-4

10-3

10-2

10-1

100

||e|| 2

LSLS DM

L2 error norm of the Poisson problem on a 25 element mesh,polynomial degree 5

Fig. 26 The L2-error of the conventional least-squares solution (red) and the solution obtained bydirect minimization (green)

approximating the solution up to a κ of O(10−11). Note that the solution is indepen-dent of κ .

Table 5 shows the condition number of the conventional least-squares and DirectMinimization approach versus polynomial degree in the spectral element method.One observes that the condition number of Direct Minimization is approximately thesquare root of the condition number associated with the conventional least-squaresformulation.

218


Condition numberN DM LS

2 5.934 35.2153 11.914 142.0564 20.228 409.3695 30.646 936.737

Table 5 Comparison of the condition numbers obtained from Direct Minimization (DM) and theconventional least-squares method (LS) as a function of the polynomial degree

Condition numberP = 2 P = 4

K DM LS DM LS

4 4.430 19.635 13.394 197.6359 5.935 35.222 20.230 409.635

16 7.688 59.111 27.060 732.63525 9.534 90.891 33.879 1147.786

Table 6 Condition number as a function of the number of elements for Direct Minimization (DM)and the conventional least-squares method (LS)

Table 6 shows the growth of the condition number as a function of the num-ber of elements for two polynomial degrees. One observes that, especially for highorder methods which employ much higher polynomial degrees than N = 4, the dif-ference in condition number between the conventional least-squares method andDirect Minimization grows very fast.

7.3 Application of LSQSEM to viscoelastic fluids

Inspired by the success of the simulation of a Newtonian flow around cylindersattempts were undertaken to solve the flow of a viscoelastic fluid around a cylinderin a channel. The model that was used was the so called Upper-Convected Maxwell(UCM) model. The UCM model is not the most realistic model for viscoelasticflows, but it is the simplest one in terms of number of physical parameters. However,this model is very hard to solve numerically due to its conditional well-posedness,see for instance [35, 39], and therefore is a very good test problem for numericalschemes.

This problem was solved using a discontinuous least-squares formulation, [36].Furthermore, Direct Minimization was used, see Subsection 7.2.

The Upper Convected Maxwell model is given by conservation of mass for anincompressible flow

∇ ·u = 0 in Ω , (120)

where u denotes the velocity vector field.

219


Conservation of momentum in the Stokes limit yields

∇ · (pI− τ) = 0 in Ω , (121)

where p denotes the pressure field, I the unit tensor in Rd , d = dim(Ω) and τ is the

extra-stress tensor.The constitutive equation which relates the extra-stress tensor to the velocity field

is given by

λ∇τ + τ = 2µd , (122)

where λ is the relaxation time and µ the polymeric viscosity of the fluid,∇· denotes

the upper convected derivative defined as

∇A =

∂A∂ t

+(u,∇)A−Lτ− τLT, (123)

where (L)i j = ∂ui/∂x j and 2d = L + LT . The UCM model (122) describes the factthat the extra-stress does not instantaneously equal the rate of deformation of theflow, but is also convected and deformed along the particle paths as expressed by(123). When the relaxation time λ = 0, the stress components are no longer con-vected along the particle paths and Newtonian Stokes flow is retrieved.

Consider the flow past a cylinder placed at the centerline of a channel of width4R, where R denotes the radius of the cylinder. The computational domain equalsthe domain used by Alves, Pinho and Oliveira, [2]. At inflow, 19R upstream of thecylinder, a fully developed Poiseuille flow is prescribed for velocity and extra-stresscomponents. The downstream length is taken to be 59R. The number of spectralelements equals K = 16 and the polynomial degree in each element has been setto N = 16 for all variables. The topology of the grid and the Gauss-Lobatto gridnear the cylinder are shown in Fig. 27. Note the small spectral elements aroundthe cylinder and in the wake of the cylinder. Especially near the rear stagnationpoint a very small spectral element is placed to capture the high stress-gradients. Inorder to compare the results obtained with D-LSQSEM-DM the non-dimensionaldrag coefficient on the cylinder is compared with results reported in [2]. The dragcoefficient is defined as

Cd :=1

µU

∫

surf. cyl(τ− pI) ·nx dS , (124)

where nx is the x-component of the outward unit normal at the cylinder and U isthe bulk velocity. The influence of elasticity in the flow is denoted by the Deborahnumber, De

De =λUR

. (125)

Fig. 28 graphically displays the drag coefficients as a function of De. From theresults presented above it can be concluded that D-LSQSEM-DM is capable of pro-ducing drag coefficients in agreement with those reported in [2]. However, agree-

220


Fig. 27 Topology of the spectral elements in the vicinity of the cylinder (top figure) and the Gauss-Lobatto grid for a polynomial degree N = 16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1De

85

90

95

100

105

110

115

120

125

130

135

140

CD

M60 SMARTM120 SMARText SMARTD-LSQSEM16

Fig. 28 Non-dimensional drag obtained with D-LSQSEM-DM with a polynomial degree N = 16compared to the finite volume results using SMART discretization reported in [2]

ment of integral quantities does not necessarily imply pointwise agreement. There-fore the contour plot of the extra-stress component in the xx-direction is comparedwith Fig. 16 taken from [2] in Fig. 29.

This comparison demonstrates that no stabilization terms are required in the leastsquares method to produce smooth and converged solutions; i.e. no oscillations arepresent. Both contour plots display a similar pattern. The highest value of the τxx-component obtained in the D-LSQSEM-DM calculation equals 128.96 at the cylin-der. This value is in agreement with the stress levels reported in [2] and later con-

221


0

5

-0.3-0.2

0

1

35

-0.3

-0.3

0

15

30

135

10

10

20 13

20

30

0

1

3

5

1020

30

1

3 50 2050

1

1

3

5

5

10

5

10

10

1015

15

10

10

15

30

0.5 2 10 30

50100 120

0

-0.2

1

0

-0.3

35

10

1

3

510

3

100

120

10

2030

3

20 105

50

3

30

20

3

2

1

0.5

53

2

5

10

15

30

50

Fig. 29 Contour lines of the τxx-component of the extra-stress tensor at De = 0.9 and detail of thecontours just behind the cylinder (top Figures) and comparison of contours of the normal stresses(τxx) near the cylinder predicted with M60 (dashed line) and M120 (solid line), at De = 0.9. Takenfrom [2] (lower Figure)

firmed in [1]. For further particulars on the use of LSQSEM for viscoelastic flowproblems see [34].

222


8 Further reading

There is a vast amount of literature on weak formulations and finite element methodsranging from very applied to the mathematical theory of variational formulations.We refer to the books by Brenner and Scott, [13] and Ern and Guermond, [31]for the mathematical theory of finite element methods. The two main books on theleast-squares finite element method are the books by Jiang, [51] and by Bochev andGunzburger, [9].

For the use of higher order/spectral method in fluid dynamics the books byCanuto et al., [15], Schwab, [77] and Karniadakis and Sherwin, [52] are excellentintroductions.

A simple introduction in the least-squares spectral element method can be foundin the VKI lecture series, [37].

Heinrichs, [42, 43, 44, 45, 46, 47], developed least-squares spectral collocationschemes for fluid flow applications. These methods are not directly based on norm-equivalence. Direct Minimization has shown that using the least-squares weak for-mulation together with Gauss-Lobatto integration can be interpreted as a weightedcollocation scheme, thus providing a potential theoretical framework for the least-squares spectral collocation schemes.

Dorao and Jakobsen, [23, 24, 25, 26, 27, 28, 29], applied LSQSEM successfullyto population balance equations to model multi-phase flows in chemical engineer-ing.

Pontaza and Reddy, [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67], applied LSQSEMto solid mechanics and fluid flow problems.

When only low order finite elements are combined with the least-squares formu-lation – the so-called least-squares finite element method (LSFEM) – the numberof scientific papers and applications is growing at an ever increasing rate signifyingthe renewed interest in the use of the least-squares formulation in engineering.

References

1. Afonso, A., Alves, M.A., Pinho, F.T., Oliveira, P.J.: Uniform flow of viscoelastic fluids past aconfined falling cylinder. Rheol. Acta, 47, pp. 325-348, (2008)

2. Alves, M.A., Pinho, F.T., Oliveira, P.J.: The flow of viscoelastic fluids past a cylinder: finite-volume high-resolution methods. J. Non-Newtonian Fluid Mech., 97, pp. 207-232, (2001)

3. Aziz, A.K., Kellogg, R.B., Stephens, A.B.: Least-squares methods for elliptic problems.Math. Comp., 44, pp. 53-77, (1985)

4. Berndt, M., Manteuffel, T.A., McCormick, S.F.: Local error estimation and adaptive refine-ment for first-order system least squares (FOSLS). Electron. Trans. Numer. Anal., 6, pp. 35-43, (1997)

5. Bjorck, A.: Least-squares methods. In: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of NumericalAnalysis, Solution of Equations in R

n, Part I, Vol. I, Elsevier, North Holland, Amsterdam, pp.466-647, 1990.

6. Bjorck, A.: The calculation of linear least squares problems. Acta Numer., 13, pp.1-53, (2004)7. Bochev, P.B., Gunzburger, M.D.: Accuracy of least-squares methods for the Navier-Stokes

equations. Comput. Fluids, 22, pp. 549-563, (1993)

223


8. Bochev, P.B., Gunzburger, M.D.: Analysis of least-squares finite element methods for theStokes equations. Math. Comp., 63, pp. 479-506, (1994).

9. Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods. Springer Verlag.Applied Mathematical Series, Vol. 166, 2008

10. Bramble, J.H., Schatz, A.: Rayleigh-Ritz-Galerkin methods for Dirichlet’s problem using sub-spaces without boundary conditions. Comm. Pure Appl. Math., 23, pp. 653-675, (1970)

11. Bramble, J.H., Schatz, A.: Least-squares for 2mth-order elliptic boundary-value problems.Math. Comp., 25, pp. 1-32, (1971)

12. Bramble, J.H., Lazarov, R.D., Pasciak, J.E.: Least-squares for second order elliptic problems.Computer Methods in Appl. Mech. Engrg. 152, pp. 195-210, (1998)

13. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. SpringerVerlag, New York, 1994.

14. Cai, Z., Lazarov, R., Manteuffel, T.A., McCormick, S.F.: First-order system least-squares forsecond-order partial differential equations: Part I. SIAM J. Numer. Anal., 31(6), pp. 1785-1799, (1994)

15. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluids Dynamics.Springer Verlag. 1988

16. Carey, G.F., Jiang, B.-N.: Least-squares finite element method and preconditioned conjugategradient solution. Int. J. Numer. Methods Eng., 24, pp. 1283-1296, (1987)

17. Coutanceau, M., Bouard, R.: Experimental determination of the main features of the viscousflow in the wake of a circular cylinder in uniform translation, Part I. Journal of Fluid Mechan-ics, 79, (1977)

18. De Maerschalck, B., Gerritsma, M.I.: The use of Chebyshev polynomials in the space-timeleast-squares spectral element method. Numerical Algorithms, 38(1-3), pp. 155-172, (2005)

19. De Maerschalck, B., Gerritsma, M.I.: Higher-order Gauss-Lobatto integration for non-linearhyperbolic equations. Journal of Scientific Computing. 27(1-3), pp. 201-214, (2006)

20. De Maerschalck, B., Gerritsma, M.I., Proot, M.M.J.: Space-time least-squares spectral ele-ment methods for convection-dominated unsteady flows. AIAA Journal. 44(3), pp. 558-565,(2006).

21. De Maerschalck, B., Gerritsma, M.I.: Least-squares spectral element method for non-linearhyperbolic differential equations. Journal of Computational and Applied Mathematics. 215,pp. 357-367, (2008).

22. Deville, M.O., Fischer, P.F., Mund, E.H.: High Order Methods for Incompressible Fluid Flow.Cambridge University Press, 2002

23. Dorao, C.A., Jakobsen, H.A.: A least-squares method for the solution of population balanceproblems. Computers and Chemical Engineering, 30,(3), pp. 535-547, (2005)

24. Dorao, C.A., Jakobsen, H.A.: Application of the least-squares method for solving populationbalance problems in Rd+1. Chemical Engineering Science, 61(15), pp. 5070-5081, (2006)

25. Dorao, C.A., Jakobsen, H.A.: Numerical calculation of moments of the population balanceequation. Journal of Computational and Applied Mathematics, 196, pp. 619-633, (2006)

26. Dorao, C.A., Jakobsen, H.A.: A parallel time-space least-squares spectral element solver forincompressible flow problems. Applied Mathematics and Computation, 185(23), pp. 45-58,(2007)

27. Dorao, C.A., Jakobsen, H.A.: Least-squares spectral method for solving advective populationbalance problems. Journal of Computational and Applied Mathematics, 201(1), pp. 247-257,(2007)

28. Dorao, C.A., Jakobsen, H.A.: hp-Adaptive least-squares spectral element method for popula-tion balance equations. Applied Numerical Mathematics, 58(5), pp. 563-576, (2008)

29. Dorao, C.A., Fernandino, M.: Simulation of transients in natural gas pipelines. Mathematicsand Computers in Simulation. To appear.

30. Dziskariani, A.V.: The least-squares and Bubnov-Galerkin methods. Z. Vycisl. Mat. i. Mat.Fiz., 8, pp. 1110-1116, (1968)

31. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer Verlag, NewYork, 2004.

224


32. Galerkin, B.G.: Series solution of some problems in elastic equilibrium of rods and plates.Vestn. Inzh. Tech., 19, pp. 897-908, (1915)

33. Galvao, A., Gerritsma, M.I., De Maerschalck, B.: hp-Adaptive least squares spectral elementmethod for hyperbolic partial differential equations. Journal of Computational and AppliedMathematics. 215(2). pp. 409-418, (2008).

34. Gerritsma, M.I.: Direct minimization of the discontinuous least-squares spectral elementmethod for viscoelastic fluids. Journal of Scientific Computing. 27(1-3), pp. 245-256, (2006).

35. Gerritsma, M.I., Phillips, T.N.: On the use of characteristics in viscoealastic flow problems.IMA J. Appl. Math., 66, pp. 127-147, (2001)

36. Gerritsma, M.I., Proot, M.M.J.: Analysis of a discontinuous least-squares spectral elementmethod. Journal of Scientific Computing. 17(1-4), pp. 297-306, (2002).

37. Gerritsma, M.I., De Maerschalck, B.: The least-squares spectral element method. In: CFD –Higher Order Discretization Methods, VKI Lecture Series 2006-01, Von Karman Institute forFluid Dynamics. Ed. by H. Deconinck and M. Ricchiuto, 2006.

38. Gerritsma, M.I., Van der Bas, R., De Maerschalck, B., Koren, B., Deconinck, H.: Least-squares spectral element method applied to the Euler equations, International Journal forNumerical Methods in Fluids., 57, pp. 1371-1395, (2008)

39. Gerritsma, M.I., Phillips, T.N.: On the characteristics and compatibility equations for theUCM model fluid. Z. Angew. Math. Mech., 88(7), 523-539, (2008)

40. Gordon, W.J., Hall, C.A.: Transfinite element methods - blending-function interpolation overarbitrary curved element domains. Numerische Mathematik, 21(2), pp. 109-129, (1973)

41. De Groot, R.: Direct Minimization of Equation Residuals in Least-Squares hp- Finite ElementMethod – Numerical study of low Reynolds number flow around a circular cylinder. MSc-thesis, TU Delft, 2004.

42. Heinrichs, W.: Least-squares spectral collocation for discontinuous and singular perturbationproblems. Journal of Computational and Applied Mathematics, 157(2), pp. 329-345, (2003)

43. Heinrichs, W.: Least-squares spectral collocation for the Navier-Stokes equations. Journal ofScientific Computing, 21(1), pp. 81-90, (2004)

44. Heinrichs, W.: Least-squares spectral collocation with the overlapping Schwartz method forthe incompressible Navier-Stokes equations. Numerical Algorithms, 43(1), pp. 61-73, (2006)

45. Heinrichs, W.: An adaptive least-squares collocation method with triangular elements forthe incompressible Navier-Stokes equations. Journal of Engineering Mathematics, 56(3), pp.337-350, (2006)

46. Heinrichs, W.: An adaptive least-squares scheme for the Burgers’ equation. Numerical Algo-rithms, 47(1), pp. 63-80, (2007)

47. Heinrichs, W., Kattelans, T.: A direct solver for the least-squares spectral collocation systemon rectangular elements for the incompressible Navier-Stokes equations. Journal of Compu-tational Physics, 227(9), pp. 4776-4796, (2008)

48. Henderson, R.D., Karniadakis, G.E.: Unstructured spectral methods for simulation of turbu-lent flows. J. Comput. Phys., 122, pp. 191-217, (1995)

49. Hoitinga, W., De Groot, R., Kwakkel, M., Gerritsma, M.I.: Direct minimization of the least-squares spectral element functional – Part I: Direct Solver. Journal of Computational Physics,227, pp. 2411-2429, (2008).

50. Houston, P., Senior, B., Suli, E.: Sobolev regularity estimation for hp-adaptive finite elementmethods. Report NA-02-02, University of Oxford, UK, 2002.

51. Jiang, B.-N.: The Least-Squares Finite Element Method – Theory and Applications in Com-putational Fluid Dynamics and Electromagnetics. Springer Verlag, Scientific Computation,1998.

52. Karniadakis, G.E, Sherwin, S.J.: Spectral/hp Element Methods for CFD. Oxford UniversityPress, 2002.

53. Kwakkel, M.: Time-dependent flow simulations using the least-squares spectral element –Application to unsteady incompressible Navier-Stokes flows. MSc-thesis, TU Delft, 2007.

54. Liu, J.-L.: Exact a-posteriori error analysis of the least-squares finite element method. Appl.Math. Comput., 116, pp. 297-305, (2000)

225


55. Lucka, A.: The rate of convergence to zero of the residual and the error in the Bubnov-Galerkin method and the method of least-squares. In Proc. Sem. Differential and IntegralEquations, No. 1, pp. 113-122 (Russian) Akad, Nauk Ukrain. SSR Inst. Mat., Kiev, Ukraine,1969

56. Maday, Y., Mavriplis, C., Patera, A.: Non-conforming mortar element methods: application tospectral discretizations. Domain Decomposition Methods, Proc. 2nd Int. Symp., Los Angeles,Califormia, pp. 392-418, (1988)

57. Pontaza, J.P.: Least-squares variational principles and the finite element method: Theory, for-mulations, and models for solid and fluid mechanics. Finite Elements in Analysis and Design,41(7-8), pp. 703-728, (2005)

58. Pontaza, J.P.: A least-squares finite element formulation for unsteady incompressible flowswith improved velocity-pressure coupling. J. Comput. Phys. 217(2), pp. 563-588, (2006)

59. Pontaza, J.P.: A spectral element least-squares formulation for incompressible Navier-Stokesflows using triangular elements. Journal of Computational Physics, 221(2), pp. 649-665,(2007)

60. Pontaza, J.P.: A new consistent splitting scheme for incompressible Navier-Stokes flows: Aleast-squares spectral element implementation. Journal of Computational Physics, 225(2), pp.1590-1602, (2007)

61. Pontaza, J.P., Reddy J.N.: Spectral/hp least-squares finite element formulation for the Navier-Stokes equations. J. Comput. Phys., 190(2), pp. 523-549, (2003)

62. Pontaza, J.P., Reddy J.N.: Mixed plate bending elements based on least-squares formulation.International Journal for Numerical Methods in Engineering, 60(5), pp. 891-922, (2004)

63. Pontaza, J.P., Reddy J.N.: Space-time coupled spectral/hp least-squares finite element formu-lation for the incompressible Navier-Stokes equations. J. Comput. Phys., 197(20), pp. 418-459, (2004)

64. Pontaza, J.P., Reddy J.N.: Least-squares finite elements for shear-deformable shells. Com-puter Methods in Applied Mechanics and Engineering, 194(21-24), pp. 2464-2493, (2005)

65. Pontaza, J.P., Reddy J.N.: Least-squares finite element formulations for one-dimensional ra-diative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer, 95(3), pp. 387-406, (2005)

66. Pontaza, J.P., Reddy J.N.: Least-squares finite element formulations for viscous incompress-ible and compressible fluid flows. Computer Methods in Applied Mechanics and Engineering,195(19-22), pp. 2454-2494, (2006)

67. Prabhhakar, V., Pontaza, J.P., Reddy, J.N.: A collocation penalty least-squares finite elementformulation for incompressible flow. Computer Methods in Applied Mechanics and Engi-neering, 197(6-8), pp. 449-463, (2008)

68. Proot, M.M.J.: The least-squares spectral element method – Theory, Implementation and Ap-plication to Incompressible Flows. PhD thesis, TU Delft, 2003.

69. Proot, M.M.J., Gerritsma, M.I.: A least-squares spectral element formulation for the Stokesproblem. Journal of Scientific Computing. 17(1-4), pp. 285-296, (2002).

70. Proot, M.M.J., Gerritsma, M.I.: Least-squares spectral elements applied to the Stokes prob-lem. Journal of Computational Physics, 181(2), pp. 454-477, (2002)

71. Proot, M.M.J., Gerritsma, M.I.: Application of the least-squares spectral element method us-ing Chebyshev polynomials to solve the incompressible Navier-Stokes equations. NumericalAlgorithms. 38(1-3), pp. 155-172, (2005).

72. Proot, M.M.J., Gerritsma, M.I.: Mass- and momentum conservation of the least-squares spec-tral element method for the Stokes problem. Journal of Scientific Computing. 27(1-3), pp.389-401, (2006).

73. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations.Springer-Verlag, 1997.

74. Lord Rayleigh (J.W. Strut): On the theory of resonance. Trans. Roy. Soc., London, A161, pp.77-118, 1870

75. Ritz, W.: Uber eine neue Methode zur Losung gewisses Variationsprobleme der mathematis-chen Physik. J. Reine Angew. Math., 135, pp. 1-61, (1908)

226


76. Rizzi, A., Viviand, H.: Numerical Methods for the Computation of Inviscid Flows with ShockWaves. Vieweg Verlag, 1981.

77. Schwab, Ch.: p- and h-Finite Element Methods. Oxford Scientific Publications. 1998.78. Spekreijse, S.P.: Multigrid Solution of the Steady Euler Equations. Stichting Mathematisch

Centrum, Amsterdam, CWI Tracts, 46, 1988.79. Spivak, M.: Calculus on Manifolds – A modern approach to classical theorems of advanced

calculus. The Benjamin Cummings Publishing Company, 1965.80. Sucker, D., Brauer, H.: Fluiddynamik bei quer angestromten Zylindern, Warme und

Stoffubertragung, 8, pp. 149-158, (1975)81. Taneda, S.: Experimental investigation of the wake behind cylinders and plates at low

Reynolds numbers. Journal of Physical Society of Japan, 11, pp. 302-307, (1956)82. Tritton, D.J.: Experiments on the flow past a circular cylinder at low Reynolds numbers.

Journal of Fluid Mechanics, 6, pp. 547-567, (1959)83. White, F.M.: Viscous Fluid Flows. McGraw-Hill, Inc., 1991.

227

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