za

SUPG method

ureShieomt stroxin do

nvectioite modays. Slly sufare takason ichangezed m

projection stabilization (LPS) methods, [4,6,17], continuous interior

put a particular method above the others. It must be also men-tioned that most of these methods depend on at least one param-eter about which there is no unanimous agreement on its optimalchoice in practical problems [26].

A different approach is to use layer-adapted meshes. Amongthese we cite Shishkin meshes (described below) [30,36], whichhave received considerable attention in recent years [13

15,27,31,32,38,42,46]. However, it is generally acknowledged thatthe main drawback of Shishkin meshes is the difculty to design

eDu b ru cu f ; in X; 1

Here, X is a bounded domain in R , d 1;2;3, its boundary @Xe given functionsWe assumRd, i.e., the

points x 2 @X such that bx nx < 0.It is well-known if e supfjbxjjx 2 Xg (j j bein

euclidean norm) boundary layers are likely to develop@X n C, although they have different structure on C0 fx 2 @Xjbx nx 0g and C fx 2 @Xjbx nx > 0g. As al-ready mentioned, these boundary layers, when present, areresponsible of the spurious oscillations that pollute the numericalapproximations obtained with standard methods unless extremelyne meshes are used. For uniform meshes, oscillations typicallydisappear when the mesh Pclet numberq Research supported by Spanish MEC under Grant MTM2009-07849.

Comput. Methods Appl. Mech. Engrg. 256 (2013) 116

Contents lists available at

A

w.eE-mail address: bosco@etsi.us.espenalty (CIP) methods [8], or discontinuous Galerkin (DG) methods[22,34] have been introduced, to cite a few of the many techniquesproposed (see [35,37] for a survey of methods). It must be noticed,however, that computational studies (see e.g., [2,25]) nd it hard to

being the disjoint union of CD and CN , b and c arand e > 0 is a constant diffusion coefcient.C @XD, C being the inow boundary of X 0045-7825/$ - see front matter 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cma.2012.12.001e thatset of

g thealongwind/PetrovGalerkin (SUPG) method [7], also known as thestreamline diffusion nite element method (SDFEM), or theGalerkin least-squares (GALS) method [23]. More recently, local

u g1; in @XD;@u@n

g2; in @XN : 2dto strongly-consistent stabilized methods like the streamline up-

posed in the literature, from upwind methods 35 years ago [44], We consider the problem1. Introduction

The numerical solution of cowhen convection dominates is, despsearch, a challenging problem nowaor nite-difference methods typicaspurious oscillations unless meshesless for all practical purposes. The reor thin regions where solutionsstandard methods, known as stabilindiffusion problemsre than 30 years of re-tandard nite-elementfer from unphysical oren so ne that are use-s the presence of layersfast. Modication of

ethods have been pro-

them on domains with nontrivial geometries, although someworks overcoming this difculty can be found in the literature[45,27].

The method we propose, however, does not suffer from theabove indicated drawbacks: It does not depend on parametersand, although it is based on the idea of simulating a Shishkin mesh,the experiments we present show it produces excellent results ondomains with nontrivial geometries.Finite-element methodsGalerkin methodShishkin mesh simulation: A new stabilifor convectiondiffusion problemsq

Bosco Garca-ArchillaDepartamento de Matemtica Aplicada II, Universidad de Sevilla, 41092 Sevilla, Spain

a r t i c l e i n f o

Article history:Received 30 July 2012Received in revised form 6 November 2012Accepted 3 December 2012Available online 19 December 2012

Keywords:Convection-dominated problemsStabilized methods

a b s t r a c t

A new stabilization procedcoarse and ne parts of adomains with nontrivial gcan be applied to differenobtain oscillation-free appnonuniform meshes and o

Comput. Methods

journal homepage: wwtion technique

is presented. It is based on a simulation of the interaction between theshkin mesh, but can be applied on coarse and irregular meshes and onetries. The technique, which does not require adjusting any parameter,abilized and non stabilized methods. Numerical experiments show it tomations on problems with boundary and internal layers, on uniform andmains with curved boundaries.

2012 Elsevier B.V. All rights reserved.

SciVerse ScienceDirect

ppl. Mech. Engrg.

l sevier .com/locate /cma

Fig. 1 for e 108, r 4e logJ, bx f x 1, c 0 and N 9. Itis however the presence of uNauN;ui in Eq. (8) that suppresses theoscillations, as we can see in Fig. 1, where the component Uc of theGalerkin approximation on a Shishkin grid with J 2N 18 is alsoshown (discontinuous line) together with the true solution at thenodes of the coarse part of the mesh.

It is remarkable that just by adding the value

a uNauN1;uN 14to the last equation of the Galerkin method for (12) and (13) we getthe oscillation-free approximation Uc . Obviously, in order to havethe value of a we have to solve the whole system (7)(10). In thepresent paper, we introduce a technique to approximate a withoutthe need to compute the whole approximation on the Shishkin grid.In Fig. 1, the approximation computed with the estimated a isindistinguishable from Uc . Numerical experiments in the presentpaper show that, in two-dimensional problems, the oscillation-freeapproximation on a coarse mesh can be obtained by this techniqueat half the computational cost of a Shishkin grid, and a more sub-stantial gain can be expected in three-dimensional problems.

Furthermore, this technique can be extended when the grid isno part of any Shishkin grid, while, at the same time, managingto get rid of the spurious oscillations. This allows to obtain accurate

s Appl. Mech. Engrg. 256 (2013) 116Pe kbkL1X2h

2e(h being the mesh size) is of the order of 1.

Let us briey describe now the idea of the method we proposein the following simple problem:

Lu eu00x bxu0x cx f x; 0 < x < 1; 3u0 u1 0: 4In (3) we assume that b, c and f are sufciently smooth functions,and that

0 < b < minx20;1

bx; 0 6 minx20;1

cx: 5

The standard Galerkin linear nite-element method for (3) and (4)on a partition or mesh 0 x0 < x1 < < xJ 1 of 0;1 obtains acontinuous piecewise linear approximation uhx to u. As it is cus-tomary, h denotes the mesh diameter, h max16j6Jhj, wherehj xj xj1, for j 1; . . . ; J. The approximation can be expressedas Ux u1u1x uJ1uJ1x, where the ujx are the basisor hat (piecewise linear) functions taking value 1 at the node xjand 0 in the rest of the nodes of the partition (thus, Uxj uj).The values uj, j 1; . . . ; J 1, are obtained by solving the linear sys-tem of equations

auh;ui f ;ui; i 1; . . . ; J 1; 6where, a is the bilinear form associated with (3), which is given by

av;w ev 0;w0 bv 0 c;w;; being the standard inner product in L20;1,

f ; g Z 10

f xgxdx:

The Shishkin mesh with J 2N nodes is composed of two uni-form meshes with N subintervals on each side of the transitionpoint xN 1 r, where

r min 12;2be logN

;

for an adequate constant b, that is, xj j1 r=N, for j 0; . . . ;N,and xNj xN jr=N, for j 1; . . . ;N. Let us consider the coarseand ne grid parts of the Galerkin approximation given by

Ucx u1u1x uN1uN1x;Uf x uN1uN1x u2N1u2N1x;so that Uc uNuN Uf is the Galerkin approximation on the Shish-kin mesh. Since for i 1; . . . ; J 1, the support of the basis functionui is xi1; xi1, we have aUc;uNj 0 and aUf ;uj 0, forj 1; . . . ;N 1. Consequently the system (6) on the Shishkin meshcan be rewritten as

aUc;ui; f ;ui; i 1; . . . ;N 2; 7aUc;ui uNauN;ui f ;ui; i N 1; 8aUc;uN uNauN;uN aUf ;uN f ;uN; 9uNauN;ui aUf ;ui f ;ui; i N 1; . . . ;2N 1: 10

We notice that were it not for the presence of the uNauN;ui in(8), the system (7) and (8) would be the equationsaU;ui f ;ui; i 1; . . . ;N 1; 11of the Galerkin approximation U U1u1 UN1uN1 for theproblem eu00x bxu0x cx f x; 0 < x < 1 r; 12u0 u1 r 0: 13

2 B. Garca-Archilla / Comput. MethodThe Galerkin approximation U for this problem, unless eN > 1=2, islikely to have spurious oscillations of large amplitude as we show inapproximations on domains with non trivial boundaries, whereShishkin meshes may be difcult to construct. In spite of this, wecall the new technique Shishkin mesh simulation (SMS), since itwas derived, as described above, in an attempt to simulate Shish-kin grids.

We must mention, however, that in the present paper we onlyconsider the case of dominant convection, both in the analysisand in the numerical experiments. The question of how to modifythe method (if necessary) when the mesh Pclet number Pe tendsto one will be addressed elsewhere.

It is well-known that the Galerkin method is a far from idealmethod in convectiondiffusion problems. Let us also notice thatdespite the good properties of stabilized methods developed in re-cent years, the SUPG method is still considered the standard ap-proach [26]. For this reason, in the numerical experimentspresented below, we compare the new method with the SUPGmethod.

0 0.2 0.4 0.6 0.8 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Fig. 1. Galerkin approximation on a uniform mesh with N 9 (continuous line) to

the solution of (12) and (13) with e 108, r 4e log2 N, b f 1, and c 0.The Uc part of the Galerkin approximation on a Shishkin mesh with J 18 (brokenline) for same e and f. Circles are the values of the true solution on the nodes.

Remark 1. Observe that for linear elements the operator Lh

j2NdThe points inN and the set X will play a role similar to that of x

s Apcoincides with

Lhvh XJj1

ev 00h bv 0h cvhjxj1 ;xj ; 20The rest of the paper is as follows. In Section 2 we describe theSMS method in detail. In Section 3 we present a limited analysis ofthe new technique. Section 4 contains the numerical experimentsand Section 5 the conclusions.

2. The new technique: Shishkin mesh simulation

2.1. The one-dimensional case

We consider (3) and (4) satisfying (5). Given a partition0 x0 < x1 < < xJ 1, of 0;1, we denote by Xh the space ofcontinuous piecewise linear functions, and by Vh the subspace ofXh of functions taking zero values at x 0 and x 1, so that wecan express

Xh spanfu0g Vh spanfuJg:

We consider the operator Lh given by

Lhvh bv 0h cvh; vh 2 Vh;

(see also Remark 1 below). We denote by uh 2 Vh the standardGalerkin linear nite-element approximation, and by ~uh the SMSapproximation. This is found as the solution of the least-squaresproblem

min~uh2Vh ;a2R

kLh~uh fkL20;xJ1; 15

subject to the restriction

a~uh;uh auhxJ1 f ;u; uh 2 Vh: 16

Observe that since uhxJ1 0 for uh not proportional to uJ1, andrecalling how we dened a in (14), Eq. (16) is similar to 7,8), andthen, the least-squares problem (15) is the way of nding the valuea hopefully close to a.

Notice also that the restriction (16) is, in fact, a set of asmany independent restrictions as interior nodes or nodal basisfunctions in Vh. Consequently, in the optimality conditions, theremust be a Lagrange multiplier for every interior node. We gatherall this multipliers in a function zh 2 Vh whose value at everynode is that of the corresponding Lagrange multiplier. Thus,the optimality conditions of the SMS approximation can be writ-ten as the following linear problem: nd ~uh; zh 2 Vh and a 2 Rsuch that

Lh~uh; LhuhL20;xJ1 ahuh; zh f ; LhuhL20;xJ1; uh 2 Vh;17

zhxJ1 0; 18a~uh;uh auhxJ1 f ;u; uh 2 Vh; 19

where here and in the sequel ; L2I denotes the standard innerproduct in L2I.

B. Garca-Archilla / Comput. Methodthat is, L applied element by element. This expression of Lh is bettersuited to the SMS method for higher-order elements, a topic thatwill be studied elsewhere.d h J1and xJ1; xJ in the one-dimensional case. This can be seen in Fig. 2,where, shadowed in grey, we show the set Xh for a triangulation ofthe unit square for b 1;1T , with CD @X, so that C0 [ C is con-sists of the sides y 1 and x 1. We show the points in N d markedwith circles.

The approximation ~uh 2 uDh Vh, is then found by solving theleast-squares problem

min kL ~u fk 2 ; 22Remark 2. The fact that the approximation ~uh is found by solvingthe least-squares problem (15) may suggest a possible relationwith the GALS method (compare for example least-squares prob-lem (15) with that in [35, p. 327]). However, they are very differentmethods, since for the examples in Section 4 in the present paper,SUPG and GALS methods are identical (see e.g. [23]), and, as shownin Section 4, the SMS method and the SUPG method produce mark-edly different results.

2.2. The multidimensional case

We will assume that bx 0 for x 2 X, and that every charac-teristic (i.e., solution of dx=dt bx) in X enters and leaves X innite time. Also for simplicity, we will assume that the domainX in (1) and (2) has a polygonal or polyhedral boundary.

Let T h a triangulation of it, that is, a partition of the closure X ofX in n-simplices with only faces and vertices in common. For everys 2 T h, letNs be the set of its vertices, and letN h [s2T hNs bethe set of vertices of T h.

Similarly to the one-dimensional case, let Xh the space of con-tinuous piecewise linear polynomials. We express

Xh Xh Vh Xh ;where uh 0 on CD if uh 2 Vh, and for uh in Xh (resp. Xh ), ifuhx 0 for x 2 N h, then x 2 C (resp. CD n C). In the standardGalerkin method, rst an element uDh 2 Xh Xh is selected such thatthe restriction uDh jCD

to CD is a good approximation to the Dirichletdata g1 in (2). This restriction is typically the interpolant or theL2CD-projection onto the restrictions to CD of functions in Xh.Then, the Galerkin approximation uh 2 uDh Vh satisesauh;uh f ;uh ehg2;uhiCN ; 8uh 2 Vh; 21where here and in the sequel, h; iC denotes the L2 inner product onC# @X, and

av ;w erv ;rw b rv cv ;w; v ;w 2 H1X:For the new method, similarly to the one-dimensional case, we

consider

Lhvh b rvh cvh:In order to describe the new approximation, we must set up themultidimensional version of the last interval xJ1; xJ in the one-dimensional case. For this purpose we denote

C0D C [ C0 \ CD:

We consider a suitable Xh with C0D @Xh (to be specied below)

and let us denote by N d and Nd the set of vertices in @Xh nCDand their indices, respectively, that is

N d N@Xh n CD; Nd fj 2 Njxj 2 N dg;

and by RNd we will refer to the set of real vectors of the form tj .

pl. Mech. Engrg. 256 (2013) 116 3~uh2uDhVh ;t2RNd

h h L XnXh

subject to the restriction

2.3. Further extensions

4 B. Garca-Archilla / Comput. Methods Apa~uh;uh Xj2Nd

tjuhxj f ;uh ehg2;uhiCN ; uh 2 Vh: 23

The optimality conditions of this problem can be written as the fol-lowing linear problem: nd ~uh 2 uDh Vh, zh 2 Vh and t 2 RNd suchthat

Lh~uh; LhuhL2XnXh ahuh; zh Lhuh; f L2XnXh ; uh 2 Vh;24zhxj 0; j 2 Nd; 25a~uh;uh

Xj2Nd

tjuhxj f ;uh ehg2;uhiCN ; uh 2 Vh; 26

where, as in the one-dimensional case, zh is the Lagrange multiplierof restriction (23).

Let us specify now the set Xh . The obvious choice of settingXh Bh where

Bh [

s\C0D ;s;

as in Fig. 2, may lead to an unstable method if there are nodesx 2 N h interior to Bh, as it can be easily checked in a numericalexperiment. To understand why in such a case the SMS methodmay be unstable, consider the limit case e 0, b constant andc 0 (so that the bilinear form a is skew-symmetric) and considera mesh as that depicted in Fig. 3, where there is one node xi interior

Fig. 2. A triangulation of the unit square with C0 [ C marked with a thicker line,the set Xh in grey and the points of N d with circles.to Bh who has only one neighbour node xk 2 N d. Then, on the onehand, the basis function ui of the node xi satises thatahui;uh 0 for all uh 2 Vh, except when uh is a multiple of thebasis function uk of the node xk. Furthermore, since the supportof ui is contained in Bh, we have that kLhuikL2XnBh 0. Conse-

Fig. 3. A triangulation of the unit square with C0 [ C marked with a thicker lineand the points of N d with circles. In grey, the sets Bh (left plot) and Xh (right plot)for b 1;1T .In the same way that the Shishkin meshes are not restricted tothe standard Galerkin method, the new technique, originally moti-vated by Shishkin meshes, is not restricted to the Galerkin methodeither. In this section we comment on how to use it with someother methods of widespread use, such as the SUPG, GALS, LPSand CIP methods.

In these methods the approximation uh 2 uDh Vh, instead ofsatisfying (21), satises

ahuh;uh f ;uhh ehg2;uhiCN ; 8uh 2 Vh; 30where ah and ; h are mesh-dependent bilinear forms. For theSUPG method they are given by

ahvh;uh avh;uh Xs2T h

dsLjs vh; b uhs; 31

f ;uhh f ;uh Xs2T h

dsf ; b uhs; 32

where ds is and adequately chosen parameter and Ljs denotes L re-stricted to s. In the GALS method the term b ruh is replaced byLjs uh (see e.g., [35] for a full description of these methods). Theextension of the SMS technique to these methods is as follows:we solve the least-squares problem (22) but we replace the restric-tion (23) by

ah~uh;uh Xj2Nd

tjuhxj f ;uhh ehg2;uhiCN ; uh 2 Vh: 33

In the numerical experiments in the present paper, we only con-sider the SMS for the SUPG method. To distinguish it from theSMS method in the previous section, we will refer to them as Galer-quently, taking ~vh ui, zh 0, tl 0 for l k and tk aui;ukwe have a nontrivial solution of

Lh~vh; LhuhL2XnXh ahuh; zh 0; uh 2 Vh; 27

zhxj 0; j 2 Nd; 28a~vh;uh

Xj2Nd

tjuhxj 0; uh 2 Vh; 29

if we set Xh Bh. It can be easily checked that this is also the case ifthe node xi interior to Bh is connected to more that one node in N d.For e > 0, one can expect a unique solution but, as we have checkedin practice, it is a solution where ~uxi O1=e.

To avoid this situation we remove from Bh the upwind triangleof any interior point. More precisely, for every node xi 2 N h, let usdenote by sxi its upwind triangle, that issxi fs 2 T hjfxi kbg \ s ;; k! 0g:Then we dene

Xh Bh n[

xi2N h\bBhsxi0B@

1CA:If there are two upwind triangles because xj kb is and edge for ksmaller than some k0, we may select the rst one of them afterordering all the elements.

With this choice of Xh , as we will show in Section 3.2.1, the fewcases were there are nontrivial solutions of (27)(29) have an easyx from the computational point of view. We also note that aslightly different choice of Xh is taken in Example 7 in Section 4,where the vector eld b has closed integral curves.

pl. Mech. Engrg. 256 (2013) 116kin-based and SUPG-based SMS methods.We could also consider the extension to higher-order

nite-element methods. The extension seems straightforward

s Apsince it consists in replacing the approximation space Vh by piece-wise quadratics or pieceswise cubics (together with Lh in (20)).

Numerical experiments (not reported here) indicate that thisstraightforward extension does not give as good results as thepiecewise linear elements, at least for one-dimensional problems.The reason seems to be the need to redene the set Xh for high-er-order elements. This will be subject of future studies.

3. Analysis of the SMS method

3.1. The one-dimensional case

Some understanding of why the newmethod is able to suppressor, at least, dramatically reduce the spurious oscillations of stan-dard approximations can be gained by analysing the problem givenby (3) and (4) when b is a positive constant and c 0. We do thisrst for the Galerkin method.

The Galerkin method.We rst consider the limit case e 0. In this case, it is easy to

check that the Galerkin approximation satises the equations

b2uj1 uj1 fj

Z xj1xj1

f xujxdx; j 1; . . . ; J 1: 34

Thus, summing separately odd and even-numbered equations weobtain the following expressions,

u2j u0 2bXji1

f2i1; j 1; . . . ; J0 1=2; 35

u2j1 uJ0 2b

XJ01=2ij

f2i; j 1; . . . ; J0 1=2; 36

where, here and in the rest of this section

J0 J; if J is odd;J 1; if J is even:

We notice that when J is odd, the expressions in (35) and (36)

are the discrete counterparts of the problems

bux f ; 0 < x < 1; u0 0; 37bux f ; 0 < x < 1; u1 0; 38which, unless f has zero mean, have different solutions. Thus, sincefor sufciently smooth f the expressions (35) and (36) are consistentwith (37) and (38), oscillations in the numerical approximation arebound to occur whenever f is not of zero mean. For J even, it is easyto check that the Galerkin equations (34) have no solution unlessf1 f3 fJ1 0, in which case the solution is not unique. Thusfor J even the Galerkin method is not stable.

However, for J odd, the Galerkin method is stable in the follow-ing sense

kuhk1 62bkfkh; 39

where

kfkh kShk1; 40and

S2j Xji1

f2i1; S2j1 XJ01=2ij

f2i; j 1; . . . ; J0 1=2: 41

The SMS technique. Here the equations are

b

B. Garca-Archilla / Comput. Method2~uj1 ~uj1 adJ1j fj; j 1; . . . ; J 1; 42

where dJ1 denotes Diracs delta functionr being the right-hand side of (48). Thus, applying again the stabilityestimate (39), we have

jaj 6 2kfkh h

4J 1 jrj: 50

From this inequality, (49) and the fact that 2J 1P J, the stabilityestimate (46) follows.

For J even, taking uh qh in (19) and applying (45) we have

0 b~uhx f ; qh aqJ1 aqJ1 2b

XJ=2j1

f2j1;

so that

a XJ=2j1

f2j1: 51dJ1j 0; if j J 1;1; if j J 1:

As Proposition 1 below shows, the SMS method is

stable independently of the parity of the grid. To prove stabilitywe will consider the auxiliary function qh 2 Vh whose nodal valuesare

qj 1 1j

b; j 1; . . . ; J 1: 43

It is easy to check that when J is odd qh satises

aqh;uh uhxJ1; uh 2 Vh; 44whereas when J is even it satises

aqh;uh 0; uh 2 Vh: 45

Proposition 1. There exist a positive constant C such that for e 0the SMS approximation ~uh satises

k~uhk1 66b

kfkh h6Jjrj

; 46

where r f ; LhqhL20;xJ1.

Proof. Recall that the SMS approximation ~uh is part of the solution~uh;a; zh of system (17)(19). Due to (44), (45) and (18) we havethat aqh; zh 0, so that taking uh qh in (17) we haveLh~uh; LhqhL20;xJ1 f ; LhqhL20;xJ1: 47

On the other hand, we notice that Lqh 21j=hj, for x 2 xj1; xjand j 1; . . . ; J 1, so that Eq. (47) becomes

2XJ1j1

1jb ~uj ~uj1hj

2XJ1j1

1j 1hj

Z xjxj1

f xdx: 48

To prove (46) we treat the cases J odd and J even separately. ForJ odd, we notice that since the equations in (42) are those of theGalerkin method except for the last one, applying the stability esti-mate (39) we have

k~uhk1 62bkfkh jaj: 49

Also, since ~uh uh aqh, from (48) it follows that

4XJ1j1

1hj

!a 2b

XJ1j1

1j uj uj1hj

r;

pl. Mech. Engrg. 256 (2013) 116 5Also, since the equations in (42) are those of the Galerkin methodexcept for the last one, which does not appear in (35) and (36) for

where Ihu is the interpolant of u in Xh. We prove different esti-

Then, it is easy to check that

s Apmates depending on whether

max16j6J1

jhj1 hj1j Ch2; 54

holds for some constant C > 0.

Proposition 2. There exist a positive constant C such that for e 0the error eh ~uh Ihu of SMS approximation satises the followingestimates:

k~ehk1 6 Chkf 0kL10;1; 55and, if (54) holds,

k~ehk1 6 Ch2kf 0k1 kf 00kL10;1: 56

Proof. We have

b2~ej1 ~ej1 a b2u1

dJ1j sj; j 1; . . . ; J 1; 57

where

sj Z xj1xj1

f x uj 12

dx:

SinceZ xjxj1

bIhux f dx 0; j 1; . . . ; J 1;

so that b~qhx; bIhux f L20;xJ1 0, it follows thatb~qhx; b~ehxL20;xJ1 0: 58

Thus, applying (46) with f replaced by s s1; . . . ; sJ1T and r by 0,we have

k~ehk1 66bkshkh: 59

Integration by parts reveals !J even, we have that for J even (35) and (36) also hold with uh re-placed by ~uh. Thus, it follows that

k~uhk 6 2b kfkh j~uJ1j: 52

The value of ~uJ1 is then obtained from (48) by replacing ~uj bytheir values as expressed in (35) and (36), which gives

2bXJ1j1

1hj

!~uJ1 4

XJ2j1

1hj 1hj1

Sj r;

and, thus,

j~uJ1j 6 4b kfkh h

2bJ 1 jrj; 53

which, together with (52) shows that (46) also holds for J even. hOnce the stability is proved, we investigate the convergence of

the SMS method by considering the basic solution,

bux f ; u0 0;

and the error

~eh ~uh Ihu;

6 B. Garca-Archilla / Comput. Methodsj Z xj1xj1

f 0xZ xxj

12ujdy dx;kskh 6 Chkf 0kL10;1; 60or

kskh 6 maxk6J1=2

Xkj1

f 0x2j1h22j h22j1

12

XJ1=2

jkf 0x2j

h22j1 h22j12

! Ch2kf 0kL10;1 kf 00kL10;1:

61We notice that from (59) and (60) the estimate (55) follows. On theother hand, if (54) holds, then the right-hand side of (61) is Oh2, sothat (59) and (61) imply (56). h

Let us nally comment the case e > 0. In this case, Eq. (42)should be modied by adding the term

e~uj ~uj1hj1

e ~uj ~uj1hj

to the left hand side. Then, the stability estimate (46) holds but withkfkh on the right-hand side replaced by

kfkh 4Jemax16j6J1hjk~uhk1;

added to the right-hand side. Thus, we have the same stability esti-mate (46) with the factor 6=b replaced by 12=b whenever

e 0 independent of e. Since ebx1=e < e forx < 1 ej logej=b, then, whenever

hJ Pej logej

b; 63

the bounds (55) and (56) hold if (62) also holds.

Remark 3. As mentioned in Section 1, in the present paper we areconcerned with large Pclet numbers. For this reason and forsimplicity, we have not pursued conditions which allow for largervalues of e than (62). Obviously, a more detailed (and lengthy)analysis would allow to obtain more general conditions on e.

3.2. The multidimensional case

3.2.1. Uniqueness of solutionsIn this section we limit ourselves to study the possibility loosing

uniqueness in the new method when e 0 and how to overcomethis in practice. These cases are an indication of when to expectthe method to perform poorly for small e > 0. In the rest of the pa-and, further,

sj f 0xj1h2j112

f 0xj1h2j12

Z xj1xj1

f 00xZ xxj

Z yxj

12ujdzdy

!dx:

pl. Mech. Engrg. 256 (2013) 116per, we denote

bXh X nXh :

We start by characterizing the solutions of (27)(29) whene 0. As we now show, they are given by those elements ~vh 2 Vhsatisfying

kLh~vhkL2bXh 0: 64

To see this, we take uh ~vh in (27) and getkLh~vhk2

L2bXh a~vh; zh 0:But taking uh zh in (29) we havea~vh; zh

Xj2Nd

tjzhxj 0;

B. Garca-Archilla / Comput. Methods Apwhere in the last equality we have applied (28). Thus, (64) follows.Conversely, if (64) holds, this implies Lh~vh 0 on bXh. But since weare assuming that e 0, we also have that avh;wh Lhvh;whfor any vh;wh 2 Vh. Thus, if for a basis function uj we have0 a~vh;uj Lh~vh;uj, it follows that its support cannot be en-tirely inside bXh, so that xj must be in N d. Then taking zh 0 andtj a~vh;uj we have a solution of (27)(29).

We devote the rest of this section to comment on a case thatmay easily arise in practice where there is a ~vh not entirely nullin X that satises (64) (and, hence, the lost of uniqueness in theSMS method when e 0). We will also comment on what possibleremedies can be applied in practice in these cases. In the rest of thesection we assume c 0, that is, Lhwh b rwh.

Consider an example like those in Fig. 4, that is, when the inte-

rior set bX h of bXh is not connected. We may have ~vh 2 Vh not null inX and yet satisfying (64). To see why, take for example the left plotin Fig. 4 and assume that b is constant with strictly positive com-ponents for simplicity. Let xj and xk be the most downwind vertices

of the isolated triangle s of bXh. Observe that since their associatedbasis functions uj and uk have linearly independent gradients, it ispossible to nd real values a and b so that auj buk 0 butb aruj bruk 0. Setting then ~vh auj buk we have that~vh 0 but it satises (64). If b is not constant, then it is possibleto nd ~vh such that kvhkL2X 1 but kLhvhkL2bXh diams2;there may not be nontrivial solutions of (27)(29) but the method

may not be very accurate since residuals of size diams2 allow forerrors of size 1 kvhkL2X. Similar arguments show that this is alsothe case of the right plot in Fig. 4. These arguments are easily ex-tended to tetrahedra in three-dimensional domains. In Section 3.2.2we show some other cases where there are nontrivial ~vh satisfying(64). We also state general conditions to prevent them.

From the practical point of view, all those conditions are satis-ed if the meshes are designed with a strip of elements on C0D asindicated in Fig. 5. One possible way to form this strip is to take thenodes on the outow boundary (which we assume partitioned intoFig. 4. Two triangulations of the unit square with bXh of disconnected interior. Theset Xh for b 1;1T is shadowed in grey and points of N d are marked with circles.segments or faces) and displace them along the normal vector toobtain the interior nodes of the strip (see Fig. 6). Then, sides orfaces in the outow boundary are replicated in the displacednodes, so that rectangles are obtained in two-dimensional prob-lems and triangular prisms in three-dimensional ones. In thetwo-dimensional case, the triangulation of the strip is obtainedby joining two opposite vertices of the rectangles. In three dimen-sions, tetrahedra can be obtained for example by adapting to trian-gular prisms the so-called Kuhns triangulation of the cube[3,16,28]. This is a partition of the unit cube into 6 tetrahedra, allof them having the origin and the vertex of coordinates 1;1;1,as common vertices. It is a simple exercise to adapt Kuhns triangu-lation of the cube to triangular prisms.

We notice, however, that there are times in practice when onecannot choose the grid because it is part of a larger problem, orbecause it is designed to satisfy some other constraints, etc. InSection 3.2.2, we comment on a different procedure to avoidisolated components in bXh.3.2.2. Further cases of lack of uniqueness

We complete the study of the previous section on the caseswhere there are nontrivial ~vh 2 Vh satisfying (64). We assumee 0 and c 0 here as well. We start by checking that only~vh 0 satises (64) when bX h is disconnected due to an element up-wind of a node interior to Bh, as it was the case discussed in Fig. 3.Let us take that example again. We rst notice that since we areassuming that e 0, then, (64) imply that ~vh is constant along char-acteristics, and since ~vh 0 on C, it must vanish on the big com-ponent of bXh. This implies that ~vh also vanish on the mostupwind vertex xk of the isolated triangle s. It also vanish on the ver-tex on the boundary. So it is only on the remaining vertex xi where itmay not vanish. But recall that the triangle s is upwind of xi, so thatb rui 0. Thus, Lh~vh 0 on s implies that ~vh 0 on s.

However, even when bXh has a connected interior, there may benontrivial ~vh 2 Vh satisfying (64) when there are triangles (resp.tetrahedra) in bXh, downwind of Xh and with one side aligned withthe (constant) wind velocity b (resp. one face parallel to b). This isthe case of the triangles with one side plotted with a thicker line inthe grids in Fig. 7 for b 1;1T . The basis function of the verticesopposite those sides have their gradients orthogonal to b. Forexample, for the triangle s marked with a + sign in the centre plotof Fig. 7, let xi be the vertex opposite to the shadowed side on thetriangle, and xj and xk the other two vertices. Since b rui 0 in s,andui 0 on the rest of bXh, then ~vh ui satises (64). For the gridin the left plot, we may take ~vh aui buj uk for any a; b 2 R,since b r~vh 0 in the part of bXh downwind Xh and ~vh is null inthe rest of bXh. As Proposition 3 below shows, there is no nontrivial~vh 2 X satisfying (64) for the triangulation on the right-hand sideof Fig. 7.

Nevertheless, notice that as it happens with the isolated compo-nents in Fig. 4, the cases depicted in Fig. 7 cannot occur if one de-signs the mesh with a strip of elements along C0D . However, ascommented at the end of Section 3.2.1, there are times in practicewhen one cannot choose the grid so that cases like those depictedin Fig. 4 or in Fig. 7 may occur. We now comment on the possibleremedies to avoid nontrivial solutions of (27)(29). For example,

one can connect isolated components of bX h by enlarging bXh withtriangles from Xh . We have checked in practice that this may leave

large portions of C0D as boundary of bXh, and this resulted in thepresence of spurious oscillations in the computed approximations.A better alternative is to enlarge the grid by rening those

pl. Mech. Engrg. 256 (2013) 116 7elements in Xh upwind of the isolated component. In Fig. 8 weshow the result of dividing into four similar triangles (red orregular renement) the two upwind neighbours of the isolated

ated

s ApFig. 5. Triangulations with strip of elements on C0D to prevent isol

8 B. Garca-Archilla / Comput. Methodcomponents in Fig. 4 and using longest edge bisection in those tri-angles that inherit a hanging node. We see that after the rene-

ment process bX h is connected. Also, the lack of uniquenessinduced by those triangles in bXh with a side parallel to b downwindof Xh can be prevented by red-rening their upwind triangles inXh , as it can be seen in Fig. 9, where no side parallel to the windvelocity b is now downwind of Xh .

The following result states general conditions guaranteeing that(64) implies ~vh 0. Consider the set of points that are not down-wind of Xh , that is,

dbXh fx 2 bX h jfx tbjt > 0g \Xh ;g:

Fig. 6. The process to build a st

Fig. 7. Triangulations of the unit square with triangles in bXh downwind of Xh and with oin grey.

Fig. 8. The triangulations in Fig. 4 after regular renement of upwind trianglesclosest to isolated components. The new set Xh for b 1;1T is shadowed in greyand points of the new N d are marked with circles.components in bXh . The set Xh for b 1;1T is shadowed in grey.pl. Mech. Engrg. 256 (2013) 116Proposition 3. Let b be constant and c 0. Then if either dbXh bX hor for any x 2 bX h n dXh there is a path in bX h from x to a pointy 2 dXh through elements with no edge or face parallel to b, then theonly element ~vh 2 Vh satisfying (64) is ~vh 0.

Proof. If ~vh 2 Vh satises (64), then ~vh 0 on dbXh, since any ~vhsatisfying (64) is constant along the characteristics x tb which,eventually intersect C where ~vh 0. If dbXh bX h, then ~vh canonly be nonzero on Xh . But elements on X

h have vertices either

on C0D where ~vh 0 or on bXh. Thus, ~vh must be zero on every ele-ment in Xh .

rip of elements along C0D .

ne side (thicker line) parallel to the wind velocity b 1;1T . The set Xh is shadowed

Fig. 9. The triangulations in Fig. 7 after regular renement of triangles in Xhupwind of bXh The set Xh for b 1;1T is shadowed in grey and points of the newN d are marked with circles.

Suppose now that, dbXh bX h. Then, for x 2 bX h n dbXh, let c bethe path in bX h connecting x to a point y 2 dbXh, not intersectingany element with a side or face parallel to b. There is no loss ofgenerality in assuming the path to be a polygonal, and exceptmaybe from the rst and last segments, the remaining segmentsjoint the barycenters or arithmetic means of the vertices of theelements it intersects. There will be a rst element s of those

intersected by cwhich is not entirely inside bX h n dbXh. Since s hasno edge or face parallel to b, then its interior s

; is not entirely

contained in bX h n dbXh, and thus, ~vh must vanish on s. But then,the previous element is in a similar situation, with ~vh vanishing onthe side or face in common with s and this side or face not beingparallel to b, so that ~vh is zero in that element too. Repeating theargument we conclude that ~vh vanish in x. h

example, we further comment on computational costs and the

B. Garca-Archilla / Comput. Methods ApFinally, for those situations in practice where one has to workwith a mesh without a strip of elements along C0D , we now studyhow to make robust the technique of red-rening adequate trian-gles in Xh . We rst notice that it is not difcult to design examples

where red-rening even all triangles in Xh does not make bX h con-nected. However, we now argue that two red-renements are en-ough to connect any isolated component. This is done as in theproof of Proposition 3 by considering a polygonal path in X joiningthe isolated component with another component upwind of it. Thesegments of the path may be assumed to join barycenters (resp.arithmetic means of the vertices) of triangles with a common side(resp. tetrahedra with a common face). Consider a renementstrategy where all sides or edges are bisected, and apply it to allelements in Xh intersecting the path. It is not difcult to analyseall possible cases, the worst cases being those where all verticesare in C0D and all sides (resp. faces) except those two intersecting

the path are also on C0D . After two renements the elements in Xh

are restricted to be in the area or volume induced by the new ver-tices of the second renement closest to the initial vertices, as weshow in Fig. 10. Then, in the two-dimensional case, the original

path is completely embedded in bX h, and, in the three-dimensionalcase, it is on the border of bX h, so that a small change moves it intobX h.

Let us mention that in the case of tetrahedra, the so-called reg-ular renement is not unique [3], and that the arguments abovealso apply if the renement is done by successive bisection ofthe six edges of the tetrahedron using, for example, the techniquein [1], or, in the case of triangles, three applications of longest-edgebisection to the three sides of a triangle.

Fig. 10. Two and three-dimensional elements showing shadowed in grey the partwhere Xh is restricted to be after two renements where all sides and edges are

bisected. Dotted lines are path entering and leaving the element through the onlysides and faces not on the boundary of X and passing through the barycenter.4. Numerical experiments

In this section we solve (1) and (2) on different domains X andwith different forcing terms and vector elds b. In all cases we takec 0. We used Matlab 7.13.0 and the backslash command to solvelinear systems. Also, for the SUPG method, we take the streamlinediffusion parameter as suggested in formulae (5)(7) in [25], whichwe reproduce here for the convenience of the reader. More pre-cisely, in the SUPG method test functions on element s are of theform uh dsb ruh, where ds is given by

ds diams; b2jbj ; if Pes > 1; 65

ds diams; b2

4e; if Pes 6 1; 66

where Pes jbjdiams;b2e , and, if u1; . . . ;ud1 are the basis functions inelement s (taking value 1 on one of the vertices and 0 on the restof them) then

diams; b 2jbjjb ru1j jb rud1j:

Here jbj stands for the euclidean norm of the vector eld b. If b is notconstant, it is evaluated at the barycenter of element s. The Matlabcodes used in this section are available from the author on request(see also [18]).

Example 1. Simulation of Shishkin grids. Since the SMS method wasconceived as a simulation of a Shishkin grid, we now check howwell it does it. We take X 0;12, b 2;3T , Dirichlet homoge-neous boundary conditions and the forcing term f such that thesolution is

ux; y x e2x1=ey2 e3y1=e:

This example is taken from Example 5.2 in [24], but in our casec 0. On the one hand, we solve this problem with the SUPGmethod on Shishkin meshes with J 2N subdivision in eachcoordinate direction. They are formed as tensor products of one-dimensional Shishkin meshes, with values rx 2e logN andry 3=2e logJ on the x and y directions, respectively. On theother hand, we take the coarse part of the Shishkin mesh, whichis a triangulation of

Xr 0;1 rx 0;1 ry; 67

with N subdivisions in each coordinate direction, and apply the SMSmethod (both Galerkin-based and SUPG based). For N 5;10;20; . . . ;320, we compute the numerical approximations andmeasure both the computing time and the L1 errors in the interiorpoints of the coarse part of the Shishkin grid (points which areshared by the three methods) for e 104 and e 108. Resultsare shown in Fig. 11. Looking at the errors, and for both values ofe, we notice that the three methods commit roughly the same er-rors, being in this example those of the SUPG-based SMS methodslightly worse in general. In terms of computational efciency,though, both SMS methods are roughly equally efcient, and mark-edly better than the SUPG method on Shishkin grids, since althoughthe three methods commit roughly the same errors, the SMS meth-ods compute the approximations between 4 and 2 times faster thanthe SUPG on the Shishkin grid (only for N 320, the SMS methodswere only 1.2 times faster than the SUPG method). In the next

pl. Mech. Engrg. 256 (2013) 116 9structure of the linear systems to be solved to obtain the differentapproximations.

Fig. 11. Relative efciency of SMS an

10 B. Garca-Archilla / Comput. Methods Appl. Mech. Engrg. 256 (2013) 116Example 2. Comparisons on the same grid. In the previous example,it may be considered unfair to compare the new methods on anN N coarse grid with the SUPG on a 2N 2N grid, since this isbound to be more expensive. In the present example we applythe methods on the same grids (uniform with diagonals runningSouthwestNortheast). The convection-problem is the same as inthe previous example, and, as before, L1 errors are measured ininterior mesh points. The SUPG method in this example gave verypoor results, with errors above 101. For this reason, and followingsuggestions in [29], we programmed it with some crosswind diffu-sion. More precisely, if in the SUPG method test functions on ele-ment s are of the form uh dsb ruh, where ds is thestabilization parameter, we used uh dsb ruh dc@xu wherethe value of dc was set by trial and error to obtain the smallesterrors. These values were dc 0:7701, 0.8783 and 0.9365, forN 10, 20 and 40 subdivisions in each coordinate direction. Forsimilar reasons, the value of ds in formulae (5)(7) in [25], wasmultiplied by 1.57, 1.615 and 1.64 for the above-mentioned num-ber of subdivisions, respectively.

The results can be seen in Fig. 12. Even though the SMSmethodscan be up to twice as costly as the SUPG on the same grid, theerrors are, however, between 13 and 60 times smaller (SUPG-basedSMS) and 26 and 130 (Galerkin-based SMS), so that the SMSmethod is computationally much more efcient.

Let us comment here on the computational cost of the methods.Recall that the SMS approximation ~uh is obtained (together withthe Lagrange multipliers zh and the values tj, j 2 Nd as the solutionof the optimality conditions (24)(26). Let u1; . . . ;un be the nodalbasis of Vh (each ui takes value 1 on one single node of thetriangulation and 0 on the rest of them), and let A and S be then n matrices with entries

ai;j aui;uj; si;j Lui; LujL2XnX

h; 1 6 i; j 6 n;

Fig. 12. Relative efciency of SMS and SUPG (with(or ai;j ahui;uj in the case of the SUPGmethod) respectively, andlet E be the nmmatrix whose columns are those of the n n iden-tity matrix corresponding to the indexes in Nd. Notice that we mayassume m qnd1=d for some q > 0, (d being the dimension of theeuclidean space where the domain X is). Observe also that A, S, andE are typically sparse matrices. The nodal values of ~uh, the values tj,j 2 Nd and the nodal values of zh are then obtained by solving a lin-ear system whose coefcient matrix is

M S 0 AT

0 0 ET

A E 0

26643775: 68

This must compared with the SUPG method where the coefcientmatrix is A, that is, a system of order n, whereas the SMS methodis of order 2n qnd1=d However, as Fig. 12 shows, the errors inthe SMS method are so small that greatly compensate for the largecomputational cost. Also, as Fig. 11 shows, the comparison isfavourable with the Galerkin method on Shishkin meshes, wheresystems of order 2dn have to be solved.

On the other hand notice that the change of variables~zh zh, changes AT and ET in M to AT and ET , respectively,so that the coefcient matrix in the SMS method is symmetric.This allows to use methods and software for sparse symmetricindenite matrices which are generally faster than methods forgeneral sparse matrices (like those in the SUPG or Galerkinmethods). A study of the performance of the direct methodsavailable today for sparse symmetric matrices can be found in[19], where the codes MA57 [11] and PARDISO, [39,40], appearas best performers. Note however that only serial versions were

d SUPG methods in Example 1.tried in [19] and that this is an area of fast development. As foriterative methods, the fact that SMS approximation can be found

crosswind diffusion) methods in Example 2.

by solving a symmetric system allows the use of three termrecurrence methods like the MINRES method of Paige andSaunders [33] (see e.g. [5,10,12,20,41] and the references citedtherein for information on preconditioning this kind of systems).We remark that for the systems in the SMS method, Matlabsbackslash command seems to take no advantage of thesymmetry of the systems, and, thus, no advantage has resultedfrom that symmetry in the experiments reported in the presentpaper.

Example 3. Irregular grids on curved domains. We now considerdomains where it may not be easy to set up a Shishkin grid. Inparticular, we consider the domain X enclosed by the followingcurve:

ct 26 71 sin92t

402 r2

p 1 11 1

" #2 cost r cos2t2 sint r sin2t

" #;

t 2 0;2p;

where r 0:9. The curve was created by tilting 45 degrees a cen-tred trochoid and altering its size in order to have the shape de-picted in Fig. 13. In this domain we consider problem (1) and (2),with b 2;3T and constant forcing term f 1, with Dirichlethomogeneous boundary conditions.

We study the behaviour of SMS methods on irregular grids,built as follows. For a given positive integer N, we considerDelaunay triangulations with N 12 points in X, where 4N of

each point randomly on the x and y directions with a uniformdistribution on h=3;h=3. We show one of such grids forN 20 in Fig. 13.

For N 40, we generated 200 random grids and on each ofthemwe computed the SUPG and SMS approximations and their L2

error in the convective derivative in bXh, that iskb rwh 1k

L2bXh;wh being each of the three approximations. The errors on the 200grids for e 104 and e 108 are depicted in Fig. 14 (marked dif-ferently for each method). The results have been reordered so thatthose of the SUPG method appear in descending order. We see thatthe SMS methods clearly improve the errors of the SUPG method,specially so in the case of the SUPG-based SMS method. Computingthe ratios of the error of the SUPG method and each of the SMSmethods and taking the arithmetic mean, the resulting values forthe Galerkin-based and SUPG-based SMS methods are, respectively,9.64 and 30.19 for e 104 and 11.42 and 37.69 for e 108. Thatis, the SMS methods commit errors that are, on average, between 4and 33 times smaller than those of the SUPG method. For e 108we repeated these computations for growing values of N, and therations between the errors of the SUPG method and the SMS meth-od grew with N. For example, for N 320, these were 46.93 and380.41 for the Galerkin-based and SUPG-based SMS methods,respectively.

In Fig. 15, where we compare the SUPG-based SMS approxi-mation that produced the largest error (left) with that of the SUPGmethod that produced the smallest error (right). We notice the

3 (

B. Garca-Archilla / Comput. Methods Appl. Mech. Engrg. 256 (2013) 116 11them are randomly distributed on @X. If No of these are on C0D ,then, we built the strip of elements along C0D as indicated inFig. 5. The remaining points are generated by rst tting auniform grid inside X and the outow strip, its diameter h in thex and y direction being the value for which the number ofpoints is the closest to N 12 No, and then by displacing

1 0.5 0 0.5

0.6

0.4

0.2

0

0.2

0.4

Fig. 13. The domain of ExampleFig. 14. Errors in convective derivative on randotypical oscillations of the SUPG method, which are located onlynear the outow boundary but are of considerable amplitude. TheSMS method, on the contrary presents no oscillations. Similarly, inFig. 16 we show the Galerkin-based SMS approximations thatproduced the largest and smallest errors (left and right, respec-tively). Both of them compare very favourably with the best case of

1 0.5 0 0.5

0.6

0.4

0.2

0

0.2

0.4

left) and a random grid (right).m grids for e 104 (left) and 108 (right).

s Ap0.1

0.2

0.3

0.4

12 B. Garca-Archilla / Comput. Methodthe SUPG method in Fig. 15. We notice however that some smallamplitude oscillations can be observed in the worst case of theGalerkin-based SMS method. We believe that these are due to theirregularity of the grid, since making the grids less irregular (i.e.,smaller random perturbations) diminishes these small oscillationsin the worst case, whereas making them more irregular increasesthem. It is to be remarked, though, that the irregularity of the griddoes not affect the SUPG-based SMS method. Some resultsexplaining the degradation of performance of the Galerkin methodon irregular grids can be found in [9,43]. Also, further numericalexperiments with the SMS method on irregular grids can be foundin [18].

1

0

1 0.8 0.6 0.4 0.2 0 0.20.4 0.6

0

Fig. 15. The worst case of the SUPG-based SMS method

1

0

1 0.8 0.6 0.4 0.2 0 0.20.4 0.6

0

0.1

0.2

0.3

0.4

Fig. 16. The worst and best cases of t

Fig. 17. The SUPG (left) and the SM0.1

0.2

0.3

0.4

pl. Mech. Engrg. 256 (2013) 116Example 4. Parabolic layers. We solve (1) on X 0;12, withe 108, b 1;0T and f constant equal to 1 with Dirichlet homo-geneous boundary conditions. This is a well-known test case (seee.g., [25]). The solution presents an exponential layer at the out-ow boundary at x 1 and parabolic or characteristic layers alongy 0 and y 1 (see e.g., [35] for a precise denition of theseconcepts). In Fig. 17 we show the SUPG and SMS approximationson a 20 20 regular grid with SouthwestNortheast diagonals.We notice that whereas the SUPG approximation suffers from thetypical oscillations along the characteristic layers, the SMS approx-imation is free of oscillations, both in the exponential layer at x 1and along the characteristic layers.

1

0

1 0.8 0.6 0.4 0.2 0 0.20.4 0.6

0

(left) and the best case of the SUPG method (right).

1

0

1 0.8 0.6 0.4 0.2 0 0.20.4 0.6

0

0.1

0.2

0.3

0.4

he Galerkin-based SMS method.

S approximation in Example 4.

In [25], several techniques to reduce the oscillationsof the SUPG method are tested. In this example, for approxima-tions wh computed on a regular 64 64 grid, the followingquantities

osc : maxy2f1=64;...;63=64g

fwh0:5; y wh0:5;0:5g; 69

smear : maxy2f1=64;...;63=64g

fwh0:5;0:5 wh0:5; yg; 70

are computed in [25] as a measure of the oscillations and the smear-ing along the characteristic layers, desirable values being, respec-tively, between 0 and 103 and between 0 and 104. The value ofosc and smear for all the methods tested in [25] except one was al-ways larger than 104. The values in the case of SMS methods werebelow 1014. The value of osc in the SUPG method in our tests coin-cided with that in [25], 0.134 (no value of smear was given in [25]).

Similar striking contrast between methods tested in [25] andthe SMS methods can be found in the experiments on randomly-generated grids for this example in [18].

Example 5. Interior layers. This example is also taken from [25].We solve (1) on X 0;12, with e 108, b cosp=3;sinp=3T , f 0, and u g on @X where

gx; y 0; if x 1 or y 6 0:7;1; otherwise:

The solution possesses an interior layer starting at x 0 andy 0:7, and an exponential layer on x 1 and on the right part ofy 0. For SMS methods, the only way we have conceived so far todeal with interior layers is to treat them as parabolic layers. For this

to be possible, the grid has to include the characteristic curve thatstarts at the discontinuity of the boundary data (or a polygonalapproximation to it in case of curved characteristics). We will referto this characteristic curve as the layer characteristic. For grids asthose depicted on the left of Fig. 18, where the layer characteristicis not part of the grid, one possibility to make it part of it is to moveto the layer characteristic its closest points in the grid, as we showon the right plot in Fig. 18 (see also the next example for an alter-native). Once this is done, on has to set the value of the solutionalong the layer characteristic as part of Dirichlet boundary condi-tions. To do this, we integrate the reduced problem (i.e., (1) and(2) when e 0) along the layer characteristic. In the present case,this procedure gives u 0:5 on the layer characteristic.

With these provisions, we compute the approximations of theSUPG method SMS methods on a 16 16 grid. The results areshown in Fig. 19 (the SMS approximations were identical and only

a dashed line. We have seen in the previous example that interior

B. Garca-Archilla / Comput. Methods Appl. Mech. Engrg. 256 (2013) 116 13Fig. 18. Left, a 8 8 uniform grid in Example 5, with the characteristic curve of theinternal layer (dashed line). Right, the grid on the left but moving to the internallayer characteristic its closest points on the grid.Fig. 19. The SUPG and SMS (Galerkin-based) approxilayers are treated in the SMS method in the same way ascharacteristic boundary layers, so that they must be part of thegrid. Rather than moving grid points to the layer as we did in theprevious example, in the present one we will enlarge the grid withmore triangles and vertices so that the layer characteristic is part ofthe Galerkin-based one is shown). As before, the SMS methodproduces an approximation with no oscillations, in sharp contrastwith the SUPG method. Comparison with the methods tested in[25] can be found in [18], with results very similar to those of theprevious example.

Example 6. Hemker problem. Here, X 3;9 3;3 n fx2y2 61g, b 1;0T and f 0. The boundary conditions are

ux; y 0; if x 3;1; if x2 y2 1;eru n 0; elsewhere:

8>: 71This problem, which was originally proposed in [21], models a hotcolumn (the unit circle) with normalized temperature equal to 1,and the heat being transported in the direction of the wind velocityb. Thus, a boundary layer appears in the upwind part of the unit cir-cumference from the lowest to highest point, and two internal lay-ers start from these two points and spread in the direction of b.Notice also that part of the boundary is curved, a feature which isoften encountered in applications.

We are going to present results corresponding to e 108 onthe grid shown in Fig. 20, which has 932 triangles and 531 nodes.The interior layer characteristics are not part of this grid, as it canbe seen in Fig. 21, where we show one of the layer characteristic inmations on a 20 20 regular grid in Example 5.

integral curves. Nevertheless, as we now show, the results of the

2

3

4

14 B. Garca-Archilla / Comput. Methods Appl. Mech. Engrg. 256 (2013) 116it (an example is shown in Fig. 21). This may be useful in thosecases in practice where, as commented at the end of Section 3.2.1one does not have the freedom to choose or move the grid points.

Thus, we now describe a general technique to enlarge grids inorder to include an inner layer characteristic. For simplicity wedescribe it for two-dimensional problems. Let c be the inner layercharacteristic. We assume that the mesh is sufciently ne so thatc can be well approximated by a straight segment in the interior ofevery triangle s it intersects. Let s be such a triangle. If c passesthrough a vertex v, then the s is bisected by c into two triangles.These two triangles are included in the enlarged triangulation.Otherwise, c intersects two sides, and the element s is divided by cinto a triangle and a quadrilateral (see an example in Fig. 22) whichin turn is divided into four triangles. We remark that it is easy toconceive better strategies to enlarge the triangulation in order toinclude the layer characteristic (techniques that avoid long-shapedtriangles, for example), but we have chosen this one due to itssimplicity. Nevertheless, as the experiments below show, itssimplicity does not prevent it from obtaining excellent results.

Fig. 23 shows the SUPG and the SMS approximations (Galerkin-

2 0 2 4 6 8

4

3

2

1

0

1

Fig. 20. The grid in Example 6.based). Oscillations can be clearly seen in the SUPG approximation,but they are absent in the SMS approximation. This can be betterobserved in the bottom plots, where a different point of view istaken. Following [2], we measure the undershoots of an approx-imation wh as minfwhg and the overshoots as maxfwh 1g. Theover and undershoots for the SUPG method were 0.04 and 0.52,whereas for both SMS methods the overshoots where smaller than1015, and the undershoots were 0. Similar results were obtainedby doubling and multiplying by four the number of subdivisions ineach coordinate direction. The results of the SMS method comparevery favourably not only with the SUPG method, but with most ofthe methods tested in [2], which had values very similar to those ofthe SUPG method. They also compare very favourably with resultsin [17], where LPS methods present oscillations of about 5% of thejump, whereas the SMS methods of less than 1013%.

Fig. 21. Left: detail on the grid in Fig. 20 showing one of the inner layer characteristic (dthe layer characteristic. New sides are plotted with thinner lines.SMS method in the present example are as good as in the previousones, even though part of the analysis in Section 3.2 does not applyto the present case.

Observe that since the four sides of the boundary are them-selves characteristic curves, we have C0D @X, so that building Xhas described in Section 2.2 results in Xh being composed by allelements touching the boundary. In this case the SMS methodIn [18], the previous experiment is repeated with the vectoreld b changed to b cosh; sinhT , for 100 equidistant values ofh between 0;p=4, with results similar to those shown here for allcases except four, where, in order to get over and undershoots oforder 1012 it was also necessary for nodes Oh2min away frominterior layers to be allowed to move (see [18] for details).

Example 7. Double-glazing test problem [12]. This is an examplewhere the vector eld b has vortices. The domain is X 1;12,f 0, and the wind velocity is y1 x2;x1 y2, so that thecharacteristic curves are the closed curves given by

fx; yj1 x21 y2 constantg;(see Fig. 24). The Dirichlet boundary conditions are u 1 on x 1and u 0 otherwise, so that there are discontinuities in the dataat corners in x 1, y 1. These discontinuities give raise to para-bolic boundary layers which, according to [12], have a structure dif-cult to compute by asymptotic techniques.

Notice that the hypotheses stated at the beginning of Section 2.2(bx 0 and all characteristics entering and leaving the domain innite time) do not apply to vector elds with vortices or closed

Fig. 22. Detail of the enlarged grid in Fig. 21 showing a triangle of the original gridbeing divided by the layer characteristic in a triangle and a quadrilateral, which wedivide into triangles by joining the vertices with their arithmetic mean.produces an approximation equal to 0 on all nodes except on thoseon x 1, where it takes value one, and this is only correct for e 0.For e > 0, better results are obtained with the SMS method if, as weshow in Fig. 24, the set Xh is shrunk so as to correspond to that of aslightly smaller rectangular domain, that is, elements are includedin Xh if they intersect the outow boundary of 1 d;1 d2 fora small d > 0 (other possibilities to obtain better results withvector elds with vortices will be reported in future works). Withthis selection of Xh , the results of the Galerkin-based SMS methodfor e 104 can be seen in Fig. 25. Also shown in Fig. 25 is the

ashed line). Right: the same grid enlarged with more triangles and nodes to include

s ApB. Garca-Archilla / Comput. MethodSUPG approximation, where we can observe oscillations along thecharacteristic layers, especially at x 1 and y 1. In sharpcontrast, the SMS method produces a nonnegative approximation.For this value of e, in order to obtain nonnegative approximationson a regular N N mesh with the SUPG method one must takeN P 144, and N P 240 with the Galerkin method; for e 105,this is achieved for N P 682 and N P 800 with the SUPG andGalerkin methods respectively, and for e 106, neither methodwas we able to obtain nonnegative approximations with all themeshes we tried up to N 1800. In sharp contrast, with the SMSmethod, both Galerkin-based and SUPG-based, nonnegativeapproximations were obtained for N P 8 and e as small as 1014.

Fig. 23. The SUPG approximation (left) and the Galerkin-based SMS approximation (rigplots.

Fig. 24. Left: the streamlines in Example 7. Centre: the solution in Example 7 for e 0:marked with circles.

Fig. 25. The Galerkin based SMS method (left) and the SUPG mpl. Mech. Engrg. 256 (2013) 116 155. Concluding remarks

A novel stabilization technique for convectiondiffusion prob-lems has been introduced, tested and partially analysed. It can beapplied to most existing methods based on conforming piecewiselinear nite elements. It consists of, rst, adding extra values tothe residual of the method on nodes next to the outow and char-acteristic boundaries. The resulting equations are taken as arestriction on a least-squares problem on the elementwise residu-als of the convectiondiffusion operator, where elements next tothe outow and characteristic boundaries (and internal layers)are left out.

ht) on the grid depicted in Fig. 20. A different point of view is taken on the bottom

005. Right: a regular mesh, with the set Xh shadowed in grey and points of N d are

ethod on a 20 20 regular grid in Example 7 for e 104.

The method has been tried in a series of standard and nonstan-dard tests, and results seem to suggests it performs manifestly bet-ter than a good deal of the methods of choice today. The testsinclude exponential and characteristic layers, and even irregular

[18] B. Garca-Archilla, Shiskin mesh simulation: complementary experiments,typescript. arXiv:1212.4321 [math NA], 2012.

[19] N.I.M. Gould, J.A. Scott, Y. Hu, A numerical evaluation of sparse direct solversfor the solution of large sparse symmetric linear systems of equations, ACMTrans. Math. Software 33 (2007) 2. Art. 10, 32 pp.

16 B. Garca-Archilla / Comput. Methods Appl. Mech. Engrg. 256 (2013) 116grids on domains with nontrivial geometries. This is so in spite ofthe method being initially conceived as a simulation on the coarsepart of a Shishkin mesh. The tests also include interior layers andconvection with vortices, with similarly outstanding results,although numerical experiments in the present paper and in [18]suggest that further research may be needed on these topics.

Besides the practical performance shown in tests, it is to be re-marked the lack of parameters in the method (in markedly contrastwith most of existing stabilized methods today). Subject of futureresearch will be extending the method to nite elements of higherdegree, as well as the possible changes when the mesh Pclet num-ber tends to one.

Acknowledgements

The author wish to thank Prof. Martin Stynes for advice andhelpful discussions in the research summarized in this paper.

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Shishkin mesh simulation: A new stabilization technique for convectiondiffusion problems1 Introduction2 The new technique: Shishkin mesh simulation2.1 The one-dimensional case2.2 The multidimensional case2.3 Further extensions

3 Analysis of the SMS method3.1 The one-dimensional case3.2 The multidimensional case3.2.1 Uniqueness of solutions3.2.2 Further cases of lack of uniqueness

4 Numerical experiments5 Concluding remarksAcknowledgementsReferences