A Streamline{Di usion Method for Nonconforming Finite Element Approximations Applied to Convection-Di usion Problems V. John
1,
2
G. Matthies , F. Schieweck
3
and L. Tobiska
4
Institut fur Analysis und Numerik Otto-von-Guericke-Universitat Magdeburg PF 4120, D-39016 Magdeburg, Germany
We consider a nonconforming streamline-di usion nite element method for solving convection-di usion problems. The theoretical and numerical investigation for triangular and tetrahedral meshes recently given by John, Maubach and Tobiska has shown that the usual application of the SDFEM gives not a sucient stabilization. Additional parameter dependent jump terms have been proposed which preserve the same order of convergence as in the conforming case. The error analysis has been essentially based on the existence of a conforming nite element subspace of the nonconforming space. Thus, the analysis can be applied for example to the Crouzeix/Raviart element but not to the nonconforming quadrilateral elements proposed by Rannacher and Turek. In this paper, parameter free new jump terms are developed which allow to handle both the triangular and the quadrilateral case. Numerical experiments support the theoretical predictions. Key words: Convection{di usion equations, streamline{di usion nite element methods, nonconforming nite element methods, error estimates 1991 Mathematics subject classi cations: 65N30, 65N15
1 2 3 4
Email:
[email protected] Email:
[email protected], Research partly supported by DFG grant EN 278/2-1. Email:
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1 Introduction The streamline{di usion nite element method (SDFEM) for the solution of convection-di usion problems has been successfully applied in the case of conforming nite element spaces. This method was proposed rst by Hughes and Brooks [6] and applied to several classes of problems. Starting with the fundamental work by Navert [13], it was mainly analyzed by Johnson and his co-workers [5,9,10]. Nowadays, the convergence properties are well-understood in the conforming case [14,17,21,22,24]. However, in the nonconforming case there are some new e ects which will be studied in this paper. For applications from computational uid dynamics, nite element methods of nonconforming type are attractive since they easily ful l the discrete version of the Babuska-Brezzi condition. Another advantage of nonconforming nite elements is that the unknowns are associated with the element faces such that each degree of freedom belongs to at most two elements. This results in a cheap local communication when the method is parallelized on MIMD-machines, see e.g. [4,7,11,18]. In order to stabilize nite element discretizations in the case of moderate and higher Reynolds numbers, several upwind methods (resulting in algebraic systems with M-matrices) have been developed and analyzed [2,3,16,18{20]. However, the main drawback of upwind methods is the restricted order of the discretization error. Therefore, the SDFEM which is known to combine good stability properties with high accuracy outside interior or boundary layers seems to be a suitable alternative to upwind schemes. In [12], a nonconforming SDFEM for the solution of the stationary Navier-Stokes equations is studied but no special attention has been paid to the dependency of the error constants on the Reynolds number. For the simpli ed model of a scalar convection-di usion problem, a rst analysis of convergence properties for a class of nonconforming SDFEM, in the following called NSDFEM, re ecting the dependency of the perturbation parameter is given in [8]. If the standard SDFEM is applied, large oscillations can occur. The theoretical and numerical investigation has shown that by adding jump terms to the discretization one can preserve the same O(hk+1=2) order of L2-convergence as in the conforming case for piecewise polynomials of degree k. It is important to note that the error analysis essentially uses the fact that there exists a conforming nite element subspace of the nonconforming approximation space. Thus, the analysis in [8] cannot be applied to the nonconforming Qrot 1 quadrilateral elements [15,18] where a suitable conforming subspace is not available. In order to overcome these diculties, we develop new jump terms which allow to handle mixed meshes consisting of triangles and quadrilaterals. With such meshes, we have more exibility to create locally re ned meshes. 2
In this paper, we propose a modi ed nonconforming SDFEM discretization for the solution of the boundary value problem
?"4u + b ru + cu = f in ;
u = 0 on ? := @ :
(1)
For this modi ed discretization, theoretical and numerical investigations demonstrate that the order of convergence is identical to that of the corresponding conforming methods. The main advantage of the new method is that in contrast to [8] no additional parameters in the jump terms have to be chosen. Moreover, numerical experiments show for the new discretization reasonable convergence rates of iterative solvers which is not the case for the theoretically supported jump term parameters in [8]. The remainder of this paper is organized as follows. In Section 2, notations and the modi ed nonconforming discretization of (1) are introduced. We also describe assumptions on the nite element spaces to be ful lled. These assumptions are satis ed in particular for the nonconforming Crouzeix/Raviart element and a nonconforming quadrilateral element. In Section 3, the coerciveness of the bilinear form and the convergence properties are studied. Finally, in Section 4 several numerical experiments are presented in order to support the theoretical results.
2 Notations and Preliminaries For simplicity we restrict our representation to the two-dimensional case but the results can also be generalized to three dimensions. Let R2 be a bounded domain with polygonal boundary ?. For the data of problem (1) we assume that " is a small positive parameter and that b = (b1; b2), c and f are suciently smooth functions satisfying the assumption 1 inf c(x) ? 2 div b(x) c0 > 0 : (2) x2
Using the space V := H01( ), the standard weak formulation of (1) reads: Find u 2 V such that for all v 2 V
"(ru; rv) + (b ru + cu; v) = (f; v);
(3)
where (; ) denotes the inner product in L2( ). Note that under the assumption (2) there exists a unique solution of problem (3). Let Th be a regular decomposition of the domain into elements K 2Th which are allowed to be 3
(open) triangles or convex quadrilaterals. By hK we denote the diameter of the element K and by h the maximum of all hK for K 2Th. Furthermore, let min max hmin K denote the minimum length of the edges of K and K and K the minimum and maximum angle between neighbouring edges of K , respectively. For the discretization of (3) we consider a family fThg of decompositions with h ! 0 which is assumed to be shape regular, i.e. there are constants C1 and 0 2 (0; ) independent of h such that min max hK =hmin K C1 and 0 K K ? 0 8 K 2 Th :
(4)
For a quadrilateral element K , we denote by K the maximum angle between two opposite edges of K and formally we set K = 0 for triangular elements. In order to get a reasonable estimate for the interpolation error of the nonconforming quadrilateral nite elements we assume
K C2hK 8 K 2 Th
(5)
with a constant C2 independent of h. The assumption (5) can be realized by the mesh generation process in the following way. We start with a coarse mesh T 0 consisting of quadrilaterals and triangles. Then, for a given mesh T ` we construct the next ner mesh T `+1 by subdividing a certain set of marked elements K 2 T ` into ner elements. For the re nement pattern of a quadrilateral element we assume that it is either subdivided into four "son-elements" by connecting the midpoints of opposite edges or that it is subdivided into triangular son-elements only. For the subdivision of a triangular element only triangular son-elements are allowed. At the end of this re nement process we get for some nest level `max our nal mesh Th := T `max . With the above assumption on the re nement pattern it can be proven that (5) is satis ed [18]. The assumption (4) admits locally adapted meshes with non-degenerating elements. However, anisotropic mesh re nement is excluded. In order to explain the nite element space Vh we need some further notations and de nitions. In the following, E (K ) denotes the set of all edges of an element K 2Th, E := [K2Th E (K ) the set of all edges of Th, E? := fE 2 E : E ?g and E0 := E n E? the set of the boundary and inner edges, respectively. For a given piecewise continuous function ', the jump [j'j]E and the average AE ' on a face E 2 E are de ned by 8 > < lim '(x + t nE ) ? tlim '(x ? t nE ) if E 6 ? !+0 [j'j]E (x) := > t!+0 ; : ? tlim ' ( x ? t n ) if E ? E !+0 4
8 > < 12 tlim ' ( x + t n ) + lim ' ( x ? t n ) if E 6 ? E E t!+0 AE '(x) := > 1 !+0 ; : 2 tlim ' ( x ? t n ) if E ? E !+0 where nE is a normal unit vector on E and x 2 E . If E ? we choose the orientation of nE to be outward with respect to otherwise nE has an arbitrary but xed orientation. For an element K 2 Th, we denote by nK the unit normal vector on @K oriented outward with respect to K . For the nonconforming nite element functions vh 2 Vh , the continuity condition of conforming nite elements at the edges E 2 E is weakened to Z JE (vh) := jE j?1 [jvhj]E d = 0 8 E 2 E 8 vh 2 Vh ; (6) E
where jE j denotes the length of the edge E . Condition (6) says that the meanvalue JE (vh) of the jumps of vh on edge E is zero for all E 2 E . Now, our nite element space Vh can be de ned as 2 Vh := vh 2 L ( ) : vhjK 2 P (K ) 8K 2 Th ; JE (vh) = 0 8 E 2 E ; (7) where P (K ) := P1(K ) if K is a triangle and P (K ) := Qrot 1 (K ) for a quadrilateral element K . Herein, P1(K ) denotes the space of all linear functions on K and Qrot 1 (K ) the space of the so-called "rotated bilinear" functions (see [15]) de ned by o n ?1 ^1; x^2; x^21 ? x^22 ; Qrot 1 (K ) := q^ FK : q^ 2 span 1; x where FK : K^ ! K is the bilinear transformation between the reference element K^ = [?1; 1]2 and the element K . The degrees of freedom (nodal values) of a function vh 2 Vh are the mean-values NE (vh) at inner edges E 2 E0 de ned by Z NE (vh) := jE j?1 vhjK d for E 2 E (K ); E
where K is an (arbitrary) element having the edge E . Because of (6), the value NE (vh) does not depend on the choice of K . Thus, the nodal values NE (v) are well-de ned for all v 2 V + Vh . Note that from the conditions JE (vh) = 0 8 E 2 E? in the de nition of Vh we have the discrete boundary conditions NE (vh) = 0 8 E 2 E?. Let 'E , E 2 E0, be the nite element basis 5
functions of Vh with NE ('E ) = 1 and NE ('E ) = 0 8E 0 2 E n fE g. Then, we de ne the interpolation operator Ih : V ! Vh by X Ihv := NE (v) 'E 8 v 2 V : (8) 0
E 2E0
In the following, we denote for a subdomain ! by (; )! the inner product in L2(!) and by k km;! and j jm;! the usual norm and semi norm in the Sobolev space H m(!), respectively. Now, the new version of the NSDFEM reads: Find uh 2 Vh such that for all vh 2 Vh
ah(uh; vh) = lh(vh)
(9)
where the bilinear form ah and the linear form lh are given by
X
"(ruh; rvh)K + (b ruh + cuh; vh)K +(?"4uh + b ruh + cuh; K b rvh)K XZ b nE [juhj]E AE vh d + E 2E E XZ jb nE j[juhj]E [jvhj]E d ; + E 2E E X lh(vh) := (f; vh) + K (f; b rvh)K :
ah(uh; vh) :=
K 2Th
(10)
K 2Th
The size of the positive control parameters K will be discussed in the next section.
Remark 1 In [8], di erent weak forms of the convection term b ru have been studied. The theoretical and numerical investigations have shown that the approximation error for the convective form of the convection term behaves better than for the skew symmetrized and the fully skew symmetrized form. Therefore, we restrict our study to the convective form used in (3) and (10). Remark 2 The jump terms introduced in [8] have the form X
Z E 2 E E [juhj]E [jvhj]E d ; E
where the parameter E has to be chosen appropriately. In contrast, now we have two parts, the rst one to guarantee coerciveness of the bilinear form and
6
the second one to ensure an "{uniform estimate of the consistency error.
Remark 3 The jump terms in the modi ed NSDFEM discretization lead to
an increased number of couplings between the local basis functions, see Figure 1. Therefore, the number of non{zero entries in the system matrix increases considerably by a factor of about 2.6 for triangular meshes and 3.28 for quadrilateral meshes. In addition, the implementation of this discretization on a parallel computer becomes more dicult.
Fig. 1. Coupling of degrees of freedoms, left without, right with jump terms.
3 Error Analysis In this section, we will provide a discretization error estimate which shows the convergence of the modi ed NSDFEM (9), (10) for h ! 0 assuming an appropriate choice of the parameters K . For the error analysis, we use the mesh-dependent norm on V + Vh
jjjvjjj2 :=
X K 2Th
XZ
E 2E E
"jvj21;K + c0kvk20;K + K kb rvk20;K +
jb nE j[jvj]2E d : 7
(11)
Due to the assumption of a shape regular family of triangulations, there exists a positive constant independent of K and the triangulation Th such that the following local inverse inequalities hold
k4vhk0;K h?K1 jvhj1;K
8vh 2 Vh ; 8K 2 Th:
(12)
We set
cmax := sup jc(x)j: x2
For the control parameter K , we assume 2 ! 1 c 0 hK 0 < K min 2 ; 2 : 2 cmax "
(13)
Let us rst consider the coerciveness of the bilinear form ah on Vh .
Lemma 4 Let Vh be the nite element space de ned by (7) and the control
parameter K ful l the assumption (13). Then, the discrete bilinear form ah is coercive on Vh , i.e., ah(vh; vh) 12 jjjvhjjj2 8vh 2 Vh : (14)
PROOF. Using the de nition of ah, element wise integration by parts and [jvhwhj]E = [jvhj]E AE wh + AE vh[jwhj]E ; (15) we obtain
ah(v; v):=
X 1Z X 2 b n "jvj21;K + (c ? 21 r b; v2)K + K v d K 2Th 2 @K K 2Th K 2Th X X + K kb rvk20;K + K (?"4v + cv; b rv)K K 2Th K 2Th XZ XZ 1 2 jb nE j [jvj]2E d : b nE 2 [jv j]E d + + E 2E E E 2E E X
Taking into consideration the de nition of the jump, the third term and the rst jump term cancel. Using (2), we get
ah(v; v) jjjvjjj2 +
X K 2Th
K (?"4v + cv; b rv)K : 8
For the remaining term, we start with
X X K j(?"4v; b rv)K j K (?"4v + cv; b rv)K K 2Th K 2Th X + K j(cv; b rv)K j:
(16)
K 2Th
Using the Cauchy-Schwarz-inequality, we get for the rst term of (16):
X K 2Th
X
"K1=2k4vk0;K K1=2kb rvk0;K K 2Th X X K kb rvk20;K: "2K k4vk20;K + 14 K 2Th K 2Th
K j(?"4v; b rv)K j
In the same way, we obtain for the second term of (16):
X K 2Th
X
K1=2cmaxkvk0;K K1=2kb rvk0;K K 2Th X X K c2maxkvk20;K + 41 K kb rvk20;K: K 2Th K 2Th
K j(cv; b rv)K j
Applying the inverse inequality (12) and the assumption (13), we get "2K k4vk20;K "2K 2h?K2jvj21;K 21 "jvj21;K and K c2max c0=2: Summarizing all estimates, we obtain
1 X K (?"4v + cv; b rv)K 2 jjjvjjj2 K 2Th
which concludes the proof of (14).
In order to prove our main result on the convergence of the proposed NSDFEM we need the following approximation properties of the space Vh. For the nite element interpolation operator Ih de ned by (8), the estimates
jv ? Ihvjm;K Ch2K?m kvk2;K kv ? (Ihv)jK k0;E Ch3K=2 kvk2;K 9
8 v 2 H 2(K ) ; 8 v 2 H 2(K ) ; E 2 E (K )
(17) (18)
are satis ed for an arbitrary element K 2Th and m 2 f0; 1; 2g provided that fThg is shape regular and the assumption (5) is ful lled for the quadrilateral elements, see [8,18]. In the following, for an edge E 2 E , the operator PE : L2(E ) ! P0(E ) denotes the L2(E )-projection into the space P0 (E ) of the constant functions on E . The estimate
kv ? PE vk0;E Ch1K=2 jvj1;K
8 v 2 H 1(K ) ; E 2 E (K )
(19)
holds, where K 2Th is an element having the edge E . For triangular elements, (19) is proven in [1]. To prove (19) for a quadrilateral element, we divide it into two triangles and apply (19) to the triangle with the edge E . The following theorem shows the convergence of the new NSDFEM (9), (10).
Theorem 5 Let the assumption of Lemma 4 be ful lled. Moreover, let the solution u of (3) belong to H01 ( ) \ H 2 ( ) and uh be the solution of (9), (10).
Then, the error estimate
1 0 2 3 2 1=2 X K hK + hK kuk2;K A jjju ? uhjjj C @ K 2Th
(20)
holds with an "{independent constant C . The parameter K is de ned by
K := " + h2K + K + K?1h2K :
(21)
PROOF. The error is split into two parts by applying the triangle inequality jjju ? uhjjj jjjIhu ? uhjjj + jjju ? Ihujjj: (22) Put wh := Ihu ? uh. From the coerciveness of the bilinear form ah, we get 1 jjjI u ? u jjj2 a (I u ? u; w ) + a (u ? u ; w ) h h h h h h h 2 h Z X @u w d : = ah(Ihu ? u; wh) + " @n h K 2Th@K
(23)
K
We set w := Ihu ? u. In order to estimate the rst term of the right hand side in (23), we consider each term of ah(Ihu ? u; wh). For the rst term, we get
X X krwk0;K krwhk0;K " (rw; rwh)K " K 2Th
K 2Th
10
0 11=2 X C "1=2 @ h2K kuk22;K A jjjwhjjj: K 2Th
Using an element wise integration by parts, we obtain for the second term (b rw + cw; wh)K = (c ? Z r b; wwh)K ? (b rwh; w)K + b nK wwh d :
(24)
@K
We sum over all cells K and get the term X Z T := b nK wwh d ; K 2Th@K
(25)
which will be estimated together with the rst jump term in (10). The remaining parts of (24) are handled as usual
X X kwk0;K kwhk0;K (c ? r b; wwh)K C K 2Th K 2Th 0 11=2 X C @ h4K kuk22;K A jjjwhjjj; K 2Th X 1=2 X K kb rwhk0;K K?1=2kwk0;K ( b r w ; w ) h K K 2Th K 2Th 0 11=2 X 2 4 ? 1 C @ K hK kuk2;K A jjjwhjjj: K 2Th
The streamline-di usion term can be estimated in the following way
X (?"4w + b rw + cw; K b rwh)K K 2Th X C K1=2 "k4wk0;K + krwk0;K + kwk0;K K1=2kb rwhk0;K K 2Th X 1=2 C K (" + hK + h2K )kuk2;K K1=2kb rwhk0;K K 2Th 0 11=2 X C @ ("2K + K h2K + K h4K )kuk22;K A jjjwhjjj K 2Th
11
0 11=2 X C @ (" + K )h2K kuk22;K A jjjwhjjj: K 2Th
To get this, we used "2K C"h2K which is a part of assumption (13). Now, we estimate the second jump term of (10)
Z X jb nE j [jwj]E [jwhj]E d E 2E E
X C
[jwj]E
0;E
jb nE j1=2 [jwhj]E
0;E E 2E
2 !1=2 X
2 !1=2 X
1 = 2 C
[jwj]E 0;E
jb nE j [jwhj]E 0;E : E 2E
E 2E
Using the de nition of the jump and (18), we get
[jwj]
C h3=2kuk2;N (E) E
E 0;E
where N (E ) is the union of all cells K with E @K and hE is the length of the edge E . It follows Z X 3 2 !1=2 X jb nE j [jwj]E [jwhj]E d C hE kuk2;N (E) jjjwhjjj: E 2E E 2E E Now, we consider the term T in (25) and the rst jump term in (10). Using (15), we obtain
X X Z K 2Th E 2E (K ) E
=?
=?
XZ
E 2E E
X
E 2E
b nK wwh d +
XZ E 2E E
b nE [jwwh j]E d +
b nE AE w[jwhj]E d :
XZ
E 2E E
b nE [jwj]E AE wh d b nE [jwj]E AE wh d
This term can be estimated in the following way
X Z b nE AE w[jwhj]E d ? E 2E E XZ jb nE j1=2jAE wjjb nE j1=2j[jwhj]E j d E 2E E
12
X
AE w 0;E jb nE j1=2[jwhj]E
0;E E 2E
2 !1=2 X
2 !1=2 X
1 = 2
jb nE j [jwhj]E 0;E :
AE w 0;E C
C
E 2E
E 2E
For each term in the rst sum, we use kAE wk0;E Ch3E=2kuk2;N (E) which follows with the help of (18). Summarizing the estimates, we get
X 3 2 !1=2 X Z b nE AE w[jwhj]E d C hE kuk2;N (E) jjjwhjjj: ? E 2E E
E 2E
In order to bound the consistency error, we use (6)
XZ X Z @u " @n wh d = ? K E 2E E K 2Th @K XZ =? E 2E E
@u [jw j] d " @n h E E
! @u @u " @n ? PE @n [jwhj]E d : E E
Applying the projection estimate (19), we see that
X X Z @u " C "hK kuk2;K jwhj1;K w d h K 2Th @K @nK K 2Th X C "1=2hK kuk2;K "1=2jwhj1;K K 2Th 0 11=2 X C @ "h2K kuk22;K A jjjwhjjj K 2Th
holds. Thus, the rst term of (22) is estimated by 0 11=2 !1=2 X X 2 2 3 2 jjjIhu ? uhjjj C @ K hK kuk2;K A + C hE kuk2;N (E) ; (26) K 2Th
E 2E
with K de ned in (21). The interpolation estimates show that
jjjwjjj2 =
XZ X 2 2 2 jb nE j[jwj]2E d "jwj1;K + c0kwk0;K + K kb rwk0;K + E 2E E
K 2Th
13
C
X K 2Th
X h2K " + h2K + K kuk22;K + C h3E kuk22;N (E) E 2E
and, using " + h2K + K K , we get for the second term of (22) X X jjjwjjj2 C K h2K kuk22;K + C h3E kuk22;N (E): K 2Th
E 2E
This gives in combination with (26) and the shape regularity of the triangulation the convergence estimate (20).
Remark 6 For the choice K hK in (21), which gives K hK in the case " < hk , the error estimate (20) yields jjju ? uhjjj C h3=2kuk2; : Thus, the new NSDFEM has the same order of convergence in the jjjjjj{norm, and consequently in the L2{norm, as the standard SDFEM for conforming P1 { triangular and Q1{quadrilateral elements, respectively, [17].
4 Numerical Studies Now, we want to investigate the numerical behaviour of the discretization schemes (9), (10) for some test examples. The goals are rst to verify the order of convergence proven in Theorem 5 and that the constant in the error estimate (20) is independent of ", second to compare the results using triangular and quadrilateral elements and third to demonstrate the e ect on the stability of the discrete solution if the jump terms are omitted in the discretization schemes. In our test examples, problem (1) is solved using the domain = (0; 1)2. The error will be measured in the L2{norm k k0; and the streamline{di usion norm kvkS on V + Vh de ned by 0 11=2 2 X kvkS = @ "jvj1;K + c0kvk20;K + K kb rvk20;K A : K 2Th
The order of convergence is computed using the errors on the two nest grids. The grids are generated by uniform re nement of a given coarse grid (level 0). Here, we present results for two kinds of coarse grids, a quadrilateral and a triangular one shown in Figure 2. 14
? ?? ? ? ? ?? ? ? ? ? ? ?? Fig. 2. Quadrilateral and triangular coarse grid (level 0).
In the following tables, the new discretization (9), (10) will be referred as "new j", the discretization with the jump terms introduced in [8] using E = 1 (see Remark 2) as "old j" and the standard NSDFEM discretization without jump terms as "no j".
Example 7 Smooth polynomial solution. We take b = (3; 2); c = 2 and the right hand side f is chosen such that u(x; y) = 100(1 ? x)2x2y(1 ? 2y)(1 ? y) is the solution of (1). The streamline{di usion parameters are chosen to be K = hK 8K 2 Th. All discretizations using jump terms show the expected order of convergence of 1.5 in the streamline{di usion norm and second order convergence in the L2{norm, see Tables 1 and 2. The same order of convergence can be observed for the standard NSDFEM without jump terms on the quadrilateral mesh whereas a reduction occurs for " = 10?8 on the triangular mesh. Between the nonconforming streamline{di usion method proposed in [8] ("old j") and the new one de ned in (9), (10) ("new j"), there are often only minor di erences. However, for the discretization (9), (10), no parameters in the jump terms have to be chosen. Considering the same level, the results on the quadrilateral mesh are better than on the triangular mesh although the quadrilateral elements need less degrees of freedom. Tables 3 and 4 present results for a xed mesh size h and di erent values of the di usion parameter ". For all discretizations with jump terms and the standard NSDFEM discretization on the quadrilateral meshes, the error is independent of ". This con rms that the constant C in the error estimate (20) does not depend on ". The order of convergence for the standard NSDFEM on triangular meshes has been studied in detail in [8].
15
Table 1 Example 7, error and order of convergence in the norm k k0; .
level grid 5 4
6 4 7 4 order 4 order
" = 10?4 no j old j new j 4.702-4 3.110-4 3.134-4 2.902-4 2.931-4 2.521-4 8.154-5 7.574-5 7.641-5 7.090-5 7.132-5 6.019-5 1.710-5 1.852-5 1.902-5 1.639-5 1.634-5 1.481-5 2.254 2.032 2.006 2.122 2.126 2.022
" = 10?8 no j old j new j 2.887-3 3.211-4 3.221-4 3.046-4 3.075-4 2.587-4 9.611-4 8.123-5 8.102-5 7.874-5 7.916-5 6.323-5 3.139-4 2.047-5 2.032-5 1.991-5 1.996-5 1.559-5 1.614 1.989 1.996 1.984 1.987 2.020
Table 2 Example 7, error and order of convergence in the norm jj jjS .
level grid 5 4
6 4 7 4 order 4 order
" = 10?4 no j old j new j 2.330-2 2.428-2 2.428-2 2.066-2 2.066-2 2.065-2 8.685-3 8.690-3 8.690-3 7.306-3 7.306-3 7.304-3 3.090-3 3.093-3 3.094-3 2.586-3 2.586-3 2.585-3 1.491 1.490 1.490 1.499 1.499 1.498
16
" = 10?8 no j old j new j 2.463-2 2.427-2 2.427-2 2.065-2 2.065-2 2.064-2 8.792-3 8.682-3 8.682-3 7.299-3 7.299-3 7.296-3 3.120-3 3.087-3 3.087-3 2.580-3 2.580-3 2.579-3 1.494 1.492 1.492 1.500 1.500 1.500
Table 3 Example 7, error on level 7 for di erent " in the norm jj jj0; .
" 10?4 10?6 10?8 10?10
triangular mesh no j new j 1.710-5 1.902-5 1.500-4 2.028-5 3.139-4 2.032-5 3.191-4 2.032-5
quadrilateral mesh no j new j 1.628-5 1.481-5 1.986-5 1.557-5 1.991-5 1.559-5 1.991-5 1.559-5
Table 4 Example 7, error on level 7 for di erent " in the norm jj jjS .
" 10?4 10?6 10?8 10?10
triangular mesh no j new j 3.090-3 3.094-3 3.117-3 3.087-3 3.120-3 3.087-3 3.142-3 3.087-3
quadrilateral mesh no j new j 2.586-3 2.585-3 2.580-3 2.579-3 2.580-3 2.579-3 2.580-3 2.579-3
Example 8 Layers at the out ow part of the boundary. Let b = (2; 3) and
c = 1. The right hand side f and the Dirichlet boundary condition are chosen such that
! ! 2(x ? 1) ? x exp 3(y ? 1) " " ! + exp 2(x ? 1) +" 3(y ? 1)
u(x; y; ")= xy2 ? y2 exp
is the solution of (1), see Figure 3. The solution has boundary layers at the lines x = 1 and y = 1. The absolute value of the Dirichlet boundary condition becomes exponentially small for " ! 0. For the computations, the parameters " = 10?8 and K = 0:25hK 8K 2 Th have been used. The thickness of the boundary layers is smaller than h for all meshes. Thus, the interpolation error in the layer region reduces the order of convergence in the global L2{norm to 0:5. The discretization methods with jump terms reach this order, see Table 5. Streamline{di usion methods are known to combine good stability with high accuracy outside the layers. The latter can be seen in Table 6 where the local L2{error measured in
~ = (0; 0:75)2 is presented. 17
x 0
0.2
0.4
0.6
0.8
1 1
0.8 0.8
0.6 0.6 y 0.4
0.4
0.2 0.2
1 0.8
0 0
0.6 0.2
0.4 y
0.4 x
0.6
0.8
0.2
0
1 0
Fig. 3. Exact solution and contour-lines of Example 8. Table 5 Example 8, Error and order of convergence in the norm k k0; .
triangular mesh level no j new j 4 1.729+0 7.129-2 5 2.461+0 5.068-2 6 3.487+0 3.593-2 7 4.898+0 2.544-2 order -0.4900 0.498
quadrilateral mesh no j new j 7.728-2 7.758-2 5.466-2 5.479-2 3.865-2 3.872-2 2.733-2 2.737-2 0.500 0.500
Table 6 Example 8, local L2{error and order of convergence in the norm k k0; ~ .
triangular mesh level no j new j 4 7.030-4 7.995-5 5 4.385-5 1.919-5 6 1.091-5 4.797-6 7 2.683-6 1.199-6 order 2.023 2.000
quadrilateral mesh no j new j 6.067-5 5.662-5 1.514-5 1.416-5 3.784-6 3.539-6 9.459-7 8.848-7 2.000 2.000
Using the standard NSDFEM without jump terms ("no j") on a triangular mesh can lead to huge oscillations inside the layers, see Figure 4. In contrast, on a quadrilateral mesh the discretization "no j" remains stable, see Table 5. Here, the use of a quadrilateral mesh shows advantages. However, a theoretical 18
explanation of this phenomenon is an open problem.
50 40 30 20 10 0 -10 -20 -30 -40 -50 1
0
0.5 0.5 1
0
Fig. 4. Example 8, solution on a triangular mesh without jump terms, level 5.
Remark 9 The linear systems have been solved using CGS with ILU as pre-
conditioner or a multigrid method with ILU {smoothing, see e.g. [23]. In the numerical tests we have observed rates of convergence which were clearly below 1 for the discretizations with jump terms and the standard discretization ("no j") on quadrilateral meshes (e.g. not worse than 0.6 for the multigrid method). In contrast, using the standard NSDFEM discretization on a triangular mesh, the rate of convergence has been very close to 1 (e.g. 0.99 and worse in the multigrid method) and the number of iterations has increased with the level of re nement. Thus, the use of jump terms on triangular meshes improves both the accuracy of the discrete solution and the convergence properties of the solvers.
Remark 10 Computations using non{rectangular quadrilateral grids have
shown similar results as in the previous examples. The behaviour of the discretizations on mixed meshes consisting of triangles and quadrilaterals (e.g. appearing in adaptive re nements of quadrilateral meshes with triangular closure) is similar to that of pure triangular meshes. In particular, huge oscillations can appear in the standard NSDFEM which can be removed by using the jump terms introduced in (10).
References [1] M. Crouzeix and P.A. Raviart. Conforming and nonconforming nite element methods for solving the stationary Stokes equations. RAIRO Numer. Anal.,
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3:33{76, 1973. [2] O. Dorok. Eine stabilisierte nite Elemente Methode zur Losung der Boussinesq-Approximation der Navier-Stokes-Gleichungen. PhD thesis, Ottovon-Guericke-Universitat Magdeburg, 1995. [3] O. Dorok, W. Grambow, and L. Tobiska. Aspects of the nite element discretizations for solving the Boussinesq approximation of the Navier-Stokes equations. In F.-K. Hebeker, R. Rannacher, and G. Wittum, editors, Numerical methods for the Navier{Stokes equations. Proceedings of the International Workshop on Numerical Methods for the Navier-Stokes equations held at Heidelberg Oct 25-28, pages 50{61. Vieweg-Verlag, 1994. [4] O. Dorok, V. John, U. Risch, F. Schieweck, and L. Tobiska. Parallel nite element methods for the incompressible Navier{Stokes equations. In E.H. Hirschel, editor, Flow Simulation with High-Performance Computers II. ViewegVerlag, Notes on Numerical Fluid Mechanics vol. 52, pages 20-33, 1996. [5] K. Eriksson and C. Johnson. Adaptive streamline di usion nite element methods for stationary convection-di usion problems. Math. Comp., 60:167{ 188, 1993. [6] T.J.R. Hughes and A.N. Brooks. A multidimensional upwind scheme with no crosswind di usion. In T.J.R. Hughes, editor, Finite Element Methods for Convection Dominated Flows, volume 34 of AMD. ASME, New York, 1979. [7] V. John. Parallele Losung der inkompressiben Navier-Stokes Gleichungen auf adaptiv verfeinerten Gittern. PhD thesis, Otto-von-Guericke-Universitat Magdeburg, 1997. [8] V. John, J. Maubach, and L. Tobiska. Nonconforming streamline-di usion nite-element-methods for convection-di usion problems. Numer. Math., 1997. to appear. [9] C. Johnson and J. Saranen. Streamline di usion methods for the incompressible Euler and Navier{Stokes equations. Math. Comp., 47:1{18, 1986. [10] C. Johnson, A.H. Schatz, and L.B. Wahlbin. Crosswind smear and pointwise errors in the streamline di usion nite element methods. Math. Comp., 49(179):25{38, 1987. [11] W. Layton, J. Maubach, and P. Rabier. Robustness of an elementwise parallel nite element method for convection-di usion problems. Technical report, ICMA-93-185 Department of Mathematics and Statistics University of Pittsburgh, 1995. [12] G. Lube and L. Tobiska. A nonconforming nite element method of streamlinedi usion type for the incompressible Navier-Stokes equations. J. Comp. Math., 8(2):147{158, 1990. [13] U. Navert. A nite element method for convection-di usion problems. PhD thesis, Chalmers University of Technology Goteborg, 1982.
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[14] K. Niijima. Pointwise error estimates for a streamline di usion nite element scheme. Numer. Math., 56:707{719, 1990. [15] R. Rannacher and S. Turek. Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Di erent. Equations, 8:97{111, 1992. [16] U. Risch. An upwind nite element method for singularly perturbed elliptic problems and local estimates in the L1 -norm. Math. Model. Anal. Numer., 24(2):235{264, 1990. [17] H.-G. Roos, M. Stynes, and L. Tobiska. Numerical Methods for Singularly Perturbed Di erential Equations. Convection-Di usion and Flow Problems. Springer-Verlag, 1996. [18] F. Schieweck. Parallele Losung der stationaren inkompressiblen Navier-Stokes Gleichungen. Habilitationsschrift, Otto-von-Guericke-Universitat Magdeburg, 1997. [19] F. Schieweck and L. Tobiska. A nonconforming nite element method of upstream type applied to the stationary Navier-Stokes equations. RAIRO Numer. Anal., 23:627{647, 1989. [20] F. Schieweck and L. Tobiska. An optimal order error estimate for an upwind discretization of the Navier{Stokes equations. Numer. Methods Partial Di erent. Equations, 12:407{421, 1996. [21] L. Tobiska. Stabilized nite element methods for the Navier-Stokes problem. In J.J.H. Miller, editor, Applications of Advanced Computational Methods for Boundary and Interior Layers, pages 173{191. Boole Press, Dublin, 1993. [22] L. Tobiska and R. Verfurth. Analysis of a streamline di usion nite element method for the Stokes and Navier{Stokes equations. SIAM J. Numer. Anal., 33(1):107{127, 1996. [23] G. Wittum. On the robustness of ILU{smoothing. SIAM J. Sci. Stat. Comput., 10:699{717, 1989. [24] G. Zhou. How accurate is the streamline di usion nite element method? J. Comp. Math., 66(217):31{44, 1997.
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