Galerkin Least Squares hp-FEM for the
Stokes Problem
Galerkin Least Squares hp-FEM pour le probleme de Stokes
Dominik Schotzau, Klaus Gerdes, and Christoph Schwab
Seminar fur Angewandte Mathematik, ETH Zurich,
Ramistrasse 101, CH-8092 Zurich, Switzerland
Comptes Rendus de l'Academie des Sciences Paris, t. 326, Serie I, pp. 249-254, 1998
Abstract: A stabilized mixed hp Finite Element Method (FEM) of Galerkin Least Squares type
for the Stokes problem in polygonal domains is presented and analyzed. It is proved that for equal
order velocity and pressure spaces this method leads to exponential rates of convergence provided
that the data is piecewise analytic.
Resume: Nous etudions la version hp d'une methode d'elements nis mixte, stabilisee par une
formulation \Galerkin Least Squares", pour le probleme de Stokes dans des domaines polygonaux.
Si les donnees sont analytiques cette methode donne des taux de convergence exponentiels pour
des espaces de vitesse et de pression d'ordres polyn^omiaux identiques.
Version francaise abregee
Pour la discretisation \Galerkin" de la formulation mixte du probleme de Stokes le choix des espaces
discrets pour la vitesse et la pression doit se faire selon la condition de stabilite de Babuska-Brezzi.
Toutefois, des espaces d'ordres polyn^omiaux identiques pour la vitesse et la pression, les plus
attirants pour l'implementation numerique, ne sont pas stables et peuvent produire des modes
parasites lors de calculs numeriques. Recemment, des methodes qui evitent ces problemes de
stabilite sont apparues, et nous referons a l'article d'apercu [5] et aux references ci-incluses a
ce sujet. Ces techniques ont deja ete appliquees a une multitude de problemes, mais toutes ces
approches considerent la version h de la methode d'elements nis. Comme la solution du probleme
de Stokes est typiquement analytique, la version hp de la methode d'elements nis peut mener
a des taux de convergence exponentiels [1]. Dans cet article nous presentons la version hp d'une
methode stabilisee, basee sur une formulation \Galerkin Least Squares", et nous montrons que la
convergence exponentielle peut ^etre obtenue, si la solution exacte possede la regularite requise.
Partant de la formulation variationnelle (1) du probleme de Stokes dans un polygone lR2 , nous
introduisons des sous-espaces discrets de version hp, VN;0 pour la vitesse, et MN;0 pour la pression,
tels que MN;0 et VN;0 soient d'ordres polyn^omiaux identiques et ne satisfassent pas la condition
de Babuska-Brezzi. Notre methode stabilisee de type \Galerkin Least Squares" est denie dans
l'equation (4) (cf. [5]). Si la force appliquee f appartient a L2 (
)2 , l'equation (4) possede une
solution unique (uN ; pN ) 2 VN;0 MN;0; ceci est postule dans la Proposition 1.
En outre, si la force f est analytique dans , la solution (u; p) du probleme de Stokes est anu la solution est singuliere.
alytique dans n [M
i=1 Ai , mis a part les coins du domaine fAi g, o
Sous l'hypothese de l'analyticite des donnees, on sait que les solutions des problemes aux limites
elliptiques appartiennent a certains espaces Bl (
) qui sont normes de maniere denombrable [1, 3].
1
Pour le probleme de Stokes, l'hypothese de regularite correspondante est: u 2 B2 (
)2 , p 2 B1 (
).
Gr^ace a cette condition, nous obtenons les estimations d'erreur de la Proposition 2, ou la partition
T est partagee en T 0 = fK 2 T : K \ Ai 6= ; pour un Ai g et T 1 = T n T 0 , pour tenir compte du
comportement dierent de la solution aux coins et a l'interieur. Nous introduisons des maillages
Tn; geometriques, ou les voisinages des coins du domaine sont ranes localement, comme illustre
dans la Figure 1. Dans le Theoreme 3 des taux de convergence exponentiels sont etablis sur ces
partitions Tn; . Pour demontrer ce resultat il faut approximer des fonctions dans Bl (
) sur des
maillages geometriques. La Proposition 4 resout ce probleme d'approximation de facon analogue a
[7]. En particulier, nous demontrons que les pressions p 2 B1 (
) L2(
) peuvent ^etre interpolees
par des fonctions continues et discretes a un taux de convergence exponentiel.
Finalement, nous considerons un exemple numerique dans un domaine en forme de L, ou la version
hp de notre methode d'elements nis stabilisee manifeste la convergence exponentielle (voir Figure
2) predite par le Theoreme 3.
1 Introduction
The classical mixed FEM discretization of Stokes ow demands properly designed FE-spaces for
the velocity and pressure that satisfy the \Babuska-Brezzi" condition. But implementationally
very attractive pairings (such as equal-order elements) do not fulll this stability condition and
lead to unphysical pressure modes in computations. Recently, \stabilized mixed methods" avoiding
this stability problem have emerged (for a survey, see, e.g., [5] and the references therein). These
techniques have already been applied to a variety of problems in uid ow, elasticity and continuum
mechanics. But all these stabilized methods have been formulated and analyzed in the context
of the h-version of the FEM. Typically, the solutions of Stokesian ow are piecewise analytic and
hence the hp-version of the FEM may lead to expontial rates of convergence [1]. In this note we
present a stabilized hp-method of Galerkin Least Squares type for the Stokes problem that features
exponential convergence provided that the solution is in an appropriate countably normed space.
2 The Galerkin Least Squares method
Let lR2 be a polygonal domain. The Stokes problem for incompressible
R uid owis:
Find a velocity eld u 2 H01 (
)2 and a pressure p 2 L20(
) = p 2 L2 (
) : pdx = 0 such that
(ru; rv) (r v; p) (r u; q) = (f; v) for all (v; q) 2 H01 (
)2 L20 (
):
(1)
Here, f 2 L2 (
)2 is a given body force per unit mass and (; ) denotes the usual L2(
)-inner
product. Note that the viscosity of the ow has been set equal to 1.
Let now T be a shape regular triangulation of into disjoint and open triangles and/or parallelograms fK g. For each element K there is an ane transformation FK such that K = FK (K^ )
where K^ is the generic reference element which is either the square Q^ = (0; 1)2 or the triangle T^ = f(x; y) : 0 < y < 1 xg. We dene as usual hK = diam(K ) and associate with each
K 2 T a polynomial degree kK . The elemental polynomial degrees are stored in the degree vector
k = fkK : K 2 T g. Let jkj = maxfkK : K 2 T g. Our hp-FEM velocity space is then
n
VN = S k;1 (T )2 = u 2 H 1 (
) : ujK FK 2 S kK (K^ ) 8 K 2 T
o2
:
(2)
Here, S k (K^ ) denotes a generic polynomial space on K^ . If K^ = Q^ we choose S k (K^ ) to be the set
of polynomials of degree k in each variable, and if K^ = T^ the set of polynomials of total degree
2
k. The hp-FEM pressure space can be discontinuous or continuous, that is
n
MN = S k;0 (T ) = p 2 L2 (
) : pjK FK 2 S kK (K^ ) 8 K 2 T
o
or
MN = S k;1 (T ): (3)
The second choice is the most convenient from an implementational point of view since it involves
spaces of equal polynomial order for the pressure and the velocity and is therefore easily incorporated into any existing FE-code. Note that these equal-order elements are instable in the classical
Galerkin discretization. We dene further VN;0 = VN \ H01 (
)2 and MN;0 = MN \ L20 (
).
Our stabilized method of Galerkin Least Squares type is for a xed > 0 dened by:
Find (uN ; pN ) 2 VN;0 MN;0 such that
B (uN ; pN ; v; q) = F (v; q) for all (v; q) 2 VN;0 MN;0;
(4)
where the bilinear form B and the functional F are given by
X h2K
B (u; p; v; q) = (ru; rv) (r v; p) (r u; q) 4 ( u + rp; v + rq )K ;
K 2T kK
X h2K
F (v; q) = (f; v) 4 (f; v + rq )K :
K 2T kK
(; )K is the L2 (K )-inner product on the element K . If = 0 then the formulation (4) reduces
to the classical Galerkin discretization of (1). The bilinear form B is symmetric. The weights
h2K =kK4 in the formulation (4) are motivated by inverse inequalities (see, e.g., [2]): There exists a
constant CI > 0 such that for all u 2 VN and p 2 MN
CI
X h2K
2
2
k4 kukL2 (K ) krukL2 (
) ;
K 2T
CI
K
X h2K
k4
K 2T
K
krpk2L2 (K ) kpk2L2 (
) :
(5)
Throughout, we make the assumption that
f 2 L2 (
)2 ;
0 < CI =2
and
kK 2 8K 2 T :
(6)
Our rst proposition addresses the stability of the scheme (4). Its proof follows strongly the lines
of [5]. As a consequence, (4) has an unique solution. The dependence on k is likely suboptimal.
Proposition 1 Under assumption (6) there exists a constant C > 0 independent of , k and the
meshwidths hK such that
B (u; p; v; q)
sup
1
1 C
2
2
2
2
jkj4
(0;0)6=(v;q)2VN;0 MN;0 (kukH 1 (
) + kpkL2 (
) ) 2 (kv kH 1 (
) + kq kL2 (
) ) 2
for all (0; 0) 6= (u; p) 2 VN;0 MN;0.
3 Error analysis and exponential convergence
If the right hand side f in (1) is analytical in it follows that u and p are analytical in n [M
i=1 Ai
where fAi g denote the vertices of (cf. Figure 1). However, there are corner singularities arising
at the vertices Ai . It is wellknown [1, 3] for closely related elasticity and potential problems that
3
in this case the solutions belong to countably normed spaces Bl (
). For the Stokes problem the
corresponding Gevrey-type regularity assumption is
u 2 B2 (
)2 ;
p 2 B1 (
)
for some 2 (0; 1).
(7)
We refer to [1] for the exact denition of these spaces. In order to capture the singular behavior
of (u; p) near corners we split the mesh T into two disjoint meshes T 0 and T 1 by setting T 0 =
fK 2 T : K \ Ai 6= ; for some vertex Ai of g, T 1 = T n T 0 . Using Proposition 1 we can derive
the following error estimate.
Proposition 2 Assume (6) and (7). Then there holds
4
ku uN kH 1 (
) + kp pN kL2 (
) C jkj ku u~k2H 1 (
) + kp p~k2L2 (
)
+
h2K nk(u u~)k2 + kr(p p~)k2 o + X h2K kf + ~u rp~k2 12
L2 (K )
L2 (K )
L2 (K )
k4
k4
K 2T 0 K
K 2T 1 K
X
for any (~u; p~) 2 VN;0 MN . C is independent of , k and the meshwidths hK .
To obtain exponential rates of convergence we must study approximation properties of Bl (
)functions on geometric meshes: The basic geometric mesh n; on Q^ with n +1 layers and grading
factor 2 (0; 1) is created recursively as follows: If n = 0, n;0 = fQ^ g. Given n; for n 0,
n+1; is generated by subdividing the element K 2 n; with 0 2 K into four smaller rectangles
by dividing its sides in a : (1 ) ratio and by removing the irregular nodes. Figure 1 shows
n; for n = 3 and = 0:5. The layers 3 and 4 are marked. A geometric mesh Tn; in the polygon
is obtained by mapping the basic mesh n; from Q^ linearly to a vicinity of each convex corner
of . At reentrant corners three suitably scaled copies of n; are used (as is indicated at A2
in Figure 1). The remainder of is then subdivided with a quasi-uniform xed partition. Of
course, other geometric renement strategies (for example with hanging nodes) are imaginable
for the local renement near the corners. We consider only polynomial degrees on Tn; which
are identical in each geometric mesh patch. A degree distribution k is called linear with slope
> 0 if the polynomial degrees are layerwise constant in the geometric patches and are given by
kj := max(2; bj c) in layer j , j = 1; : : : ; n +1. In the interior of the elemental polynomial degree
is set constant to max(2; b(n + 1)c).
x^2
n;
1
A4
layer 4
geo.eps
49 34 mm
geomesh.eps
40 34 mm
layer 3
0
A3
Tn;
A2
1 x^1
A1
A5 = A0
Figure 1: The basic geometric mesh n; on Q^ and a geometric mesh Tn; on .
4
Theorem 3 Assume (6) and (7) and let Tn; be a geometric mesh on . Then there exists
0 = 0 (; ) > 0 such that for linear degree vectors k with slope 0 , kj = max(2; bj c),
j = 1; : : : ; n + 1, the discrete solution (uN ; pN ) 2 VN;0 MN;0 of (4) with VN = S k;1 (Tn; )2 ,
MN = S k;0 (Tn; ) or MN = S k;1 (Tn; ) satises
ku uN kH 1 (
) + kp pN kL2 (
) C exp( bN 13 )
where C; b > 0 are independent of N = dim(VN ) dim(MN ) and (but depend on , and ).
The detailed proof of Theorem 3 can be found in [9] and we highlight here the main steps. It is
wellknown [7] that functions u 2 B2 (
) and p 2 B1 (
) can be approximated at an exponential
rate of convergence in S k;1 (Tn; ) and S k;0 (Tn; ), respectively. Modifying the arguments of [7] the
second statement still holds true if p 2 B1 (
) is approximated continuously in S k;1 (Tn; ). Thereby,
we use a weighted version of the Poincare inequality in the union Ki = [fK : K \ Ai 6= ;g of
elements abutting on the vertex Ai , namely
p
1 Z pdx2
jK j
i
Ki
L2 (Ki )
C diam(Ki )2(1
) kpk2 1;1
H (Ki )
for all p 2 H1;1 (Ki );
where we refer again to [1] for the denition of the weighted Sobolev spaces Hk;l (Ki ). These
approximation results are summarized in the next proposition.
Proposition 4 Let l = 1; 2 and f 2 Bl (
) for 2 (0; 1). Then there exists 0 > 0 such that
for linear degree vectors k with slope 0 , kj = max(2; bj c), j = 1; : : : ; n + 1, there is an
interpolant ' 2 S k;1 (Tn; ) with
kf 'kH l
1
(
) +
1
h2K jf 'j2
H l (K ) C exp( bN 3 )
4
k
K
1
K 2Tn;
X
(8)
where C; b > 0 are independent of N = dim(S k;1 (Tn; )). jjH l (K ) denotes the usual H l (K )0 and to
seminorm. For l = 2 the interpolant ' can be chosen to be bilinear on each K 2 Tn;
1
satisfy the zero boundary conditions if additionally f 2 H0 (
). For l = 1 ' can be chosen to be
0 .
constant on each K 2 Tn;
4 Numerical Examples
We demonstrate the performance of our stabilized method in an L-shaped domain appearing, for
example, in the backward facing step ow problem. We use an exact solution which is singular at
the reentrant corner and satises (7) (cf. [10]). The implementation of the method is based on a
general and exible hp-FE code [4] for elliptic systems that can handle geometric renement with
hanging nodes. We use C 0 elements both for the velocities and the pressure. In the computations
we use a direct solver, but our hp-FE code can also be put into the domain decomposition framework
in connection with iterative solvers [8]. We refer to [6] for additional numerical results and a detailed
description of our implementation. In Figure 2 we show the relative error in the H 1 - and L2-norm
for the rst velocity component and the pressure, respectively. The results are obtained with the
uniform polynomial degree k = 8 on geometrically rened meshes with = 0:5 and = 0:1 and
demonstrate the exponential convergence as predicted by Theorem 3.
5
hp−version performance, alpha=0.1, sigma=0.5
−1
10
relative error in H1−norm/L2−norm
1st velocity comp.
pressure
−2
10
hpbild.eps
61 50 mm
−3
10
−4
10
5
10
15
20
number of degrees of freedom 1/3
25
30
Figure 2: Relative errors for rst velocity component and pressure in an L-shaped domain.
References
[1] Babuska I. and Guo B.Q., 1988/89. Regularity of the solution of elliptic problems with piecewise
analytic data I,II, SIAM J. Math. Anal., 19, 172-203, and 20, 763-781.
[2] Bernardi C. and Maday Y., 1992. Approximations spectrales de problemes aux limites elliptiques,
Masson Springer Publishers, Paris-New York.
[3] Bolley P., Dauge M. and Camus J., 1985. Regularite Gevrey pour le probleme de Dirichlet dans
des domaines a singularites coniques, Comm. in P.D.E., 10 (No. 4), 391-431.
[4] Demkowicz L., Gerdes K., Schwab C., Bajer A. and Walsh T., 1997. HP90: A general and
exible hp-FE code in Fortran 90, in preparation.
[5] Franca L.P., Hughes T.J.R. and Stenberg R., 1993. Stabilized Finite Element Methods, Incompressible Computational Fluid Dynamics: Trends and Advances, editors M.D. Gunzburger and
R.A. Nicolaides, Cambridge University Press.
[6] Gerdes K., Schotzau D., 1997. hp-FEM for incompressible uid ow - stable and stabilized,
Research Report 97-18, Seminar fur Angewandte Mathematik, ETH Zurich.
[7] Guo B.Q. and Babuska I, 1986. The hp-version of the nite element method I: The basic approximation results; and part II: General results and applications, Comp. Mech., 1, 21-41 and 203-226.
[8] Oden J.T., Patra A. and Feng Y., 1997. Parallel Domain Decomposition Solver for Adaptive hp
Finite Element Methods, SIAM J. Numer. Anal., 34, 2090-2118.
[9] Schotzau D., Doctoral Dissertation, ETH Zurich, in preparation.
[10] Verfurth R., 1996. A review of a posteriori error estimation and adaptive mesh-renement techniques, Wiley-Teubner, Chichester-Stuttgart.
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