A P4 BUBBLE ENRICHED P3 DIVERGENCE-FREE FINITE ELEMENT ON TRIANGULAR GRIDS

Transcription

1 A P4 BUBBLE ENRICHED P3 DIVERGENCE-FREE FINITE ELEMENT ON TRIANGULAR GRIDS SHANGYOU ZHANG DEDICATED TO PROFESSOR PETER MONK ON THE OCCASION OF HIS 6TH BIRTHDAY Abstract. On triangular grids, the continuous P k plus discontinuous P k mixed finite element is stable for polynomial degree k 4. When k = 3, the inf-sup condition fails and the mixed finite element converges at an order that is two orders lower than the optimal order. We enrich the continuous P 3 by adding some P 4 divergence-free bubble functions, to be exact, one P 4 divergence-free bubble function each component each edge. We show that such an enriched P 3-P 2 mixed element is infsup stable, and converges at the optimal order. Numerical tests are presented, comparing the new element with the P 4-P 3 element and the unstable P 3-P 2 element. AMS subject classifications. 65M6, 65N3, 76M, 76D7. Keywords. finite element, divergence-free element, Stokes equations, triangular grid.. Introduction It is a challenge to construct stable H conforming mixed finite elements satisfying the incompressible condition exactly, in computing the Stokes or Navier-Stokes equations. That is, the velocity is approximated by the continuous piecewise polynomials of degree k and the pressure is approximated by the discontinuous piecewise polynomials of one degree less. Here the method is truly conforming in the sense that the finite element velocity is the H projection of the true solution in a polynomial subspace. A breakthrough on the method was done by Scott and Vogelius in 985 [8, 9] that the method is stable and consequently of the optimal order of convergence on 2D triangular grids, for the P k -P k element, if k 4, a magic number. What is this magic number k in 3D? Or if there is such a magic number in 3D? The problem remains open, after it was posted explicitly for so many years [8]. Scott and Vogelius showed that the P k -P k element is not stable on general triangular grids, if k < 4. However, on special triangular grids,

2 low order elements may be stable. On Hsieh-Clough-Tocher macro-element grids, where each base triangle is split into 3 triangles by connecting the barycenter with three vertices, the P 2 -P and the P 3 -P 2 mixed elements are stable, cf. [, 7, 22]. On Powell-Sabin macro-element grids, where each triangle is split into 6 sub-triangles, even the P -P subspace element is stable [26]. When enriching the continuous P k velocity space by some rational functions, Guzman and Neilan showed the enriched P k -P k element is table for all k, [9, ]. With additional continuity constraints, Falk and Neilan showed that the P k -P k element is stable if the continuous P k velocity is also C at vertices and the discontinuous P k pressure is C at vertices, cf. [8]. In 3D, the P k -P k mixed element is stable for all k 3 on the Hsieh- Clough-Tocher macro-element tetrahedral grids where each base tetrahedron is split into 4 sub-tetrahedra by connecting the barycenter with four vertices, cf. [25]. If splitting further a base tetrahedron into 2 sub-tetrahedra connecting the barycenter with 4 vertices and 4 face-triangle barycenters, the P k -P k is stable for all k 2, cf. [29]. On the uniform tetrahedral grids, i.e., each cube is subdivided into 6 tetrahedra, the continuous P k with discontinuous P k mixed finite element is stable for all k 6, cf. [28]. With additional constraints on the finite element spaces, Neilan showed that the P k -P k element is stable for k 6 on general tetrahedral grids if the continuous velocity finite element is C 2 continuous at all vertices and also C continuous on all edges, and the discontinuous pressure finite element function is C at all vertices and C on all edges, cf. [6]. But the Scott-Vogelius problem is still open, on general tetrahedral grids. On rectangular grids, this problem is simple that Q k,k Q k,k -Q k element and its nd version is stable for all k 2, where Q k,k denotes the continuous piecewise polynomials of separated degrees k and k in its first variable and second varialbe, respectively, cf. [4, 5, 27]. The mixed finite element of continuous P 3 velocity and discotinuous P 2 pressure is not stable on general triangular grids in 2D, cf. [8, 9]. In this work, we enrich the continuous P 3 space by some divergence-free P 4 bubble functions. Such P 4 bubble functions do not provide additional approximation power, but do provide additional degrees of freedom to relax the locking problem of the divergence-free constraint. Such a finite element enrichment technique is used before, many times. For example, mentioned above, Guzman and Neilan enrich the continuous P 3 velocity by some rational bubble functions to obtain an inf-sup stable mixed finite element [9, ]. Here, instead of rational functions whose numerical integration formula are unknown we use the P 4 bubble polynomials in this work. In the low order mixed finite element methods for the linear elasticity equation, the Hdiv P k finite element space must be enriched by P d+ divergence-free bubble functions in d-dimensional space, cf. [2, 3,, 2, 3]. 2

3 2. The enriched P 3 divergence-free element In this section, we define the P 4 -enriched P 3 divergence-free finite element. Its uni-solvence is shown. We consider a model stationary Stokes problem: Find the velocity u and the pressure p on a 2D polygonal domain Ω, such that 2. u + p = f in Ω, div u = in Ω, u = on Ω. The standard variational form for 2. is: Find u H Ω2 and p L 2 Ω := L2 Ω/C = {p L 2 Ω p = } such that 2.2 au, v + bv, p = f, v v H Ω 2, bu, q = q L 2 Ω. Here H Ω2 is the subspace of the Sobolev space H Ω 2 cf. [7] with zero boundary trace, and the blinear forms are defined by au, v = u v dx, Ω bv, p = div v p dx, Ω f, v = f v dx. Ω Let T h be an initial triangulation of Ω. We refine each triangle into four congruent triangles by connecting the three mid-edge points. This way, one grid is refined to the next level grid. We denote each triangulation in the sequence of grids also by T h, where h is the grid size. Here we introduce the multigrids [23, 24], instead of general quasi-uniform grids, to avoid the technical details of nearly-singular points. For an initial triangulation, we may have a few singular points [4, 8]. Here a singular point is a point at which all edges of an triangulation fall into two crossing lines at the point. There are exactly four types of singular points, three boundary ones and one internal one, shown in Figure 2.. There is a minor constraint for the discrete pressure functions at the singular points. However, all singular points of the initial grid will stay singular and no new singular points appear, after the multigrid refinement. Let the P 4 -enriched P 3 velocity space be, for K T h, 2.3 V K = {v P 4 K 2 div v P 2 }. Here P k stands for the space of polynomials of degree k or less. We note that, for a P 2 4 vector, the divergence is a P 3 polynomial, not a P 2 polynomial. That is, under the constraint, the x 3, x 2 y, xy 2 and y 3 coefficients of the polynomial of the divergence must be zero. With these four constraints, we 3

4 s s s s Figure 2.. There boundary singular points left three and an internal singular point. would expect the dimension of V K be 2 dim P 4 4 = 26 = 2 dim P This is to be proved in Lemma 2.. But we would like to give another equivalent definition of V K. As the range of the divergence operator on P3 2 space is P 2 already, the newly added P4 2 functions would be divergencefree. That is, the above 6 dimension space is spanned by the following 6 divergence-free P4 2 polynomials: curlx 5, curlx 4 y, curlx 3 y 2 2x = 3 y 3x 2 y 2, curlx 2 y 3, curlxy 4, curly 5. Also there, in Lemma 2., the 26 degrees of freedom in V K are given by vx i, three vertex values of two components, 2.4 e i v x j ds, -th, st, 2nd moments on three edges, K v dx, -th moment on the element. The mixed element spaces are defined by, also equivalently by 2.3 and 2.2, V h = { v h CΩ 2 v h K V K K T h, and v h Ω = }, P h = {div v h v h V h }. Since Ω p h = Ω div v h = Ω u h = for any p h P h, we conclude that V h H Ω 2, P h L 2 Ω, i.e., the mixed-finite element pair is conforming. The resulting system of finite element equations for 2.2 is: Find u h V h and p h P h such that 2.7 au h, v + bv, p h = f, v v V h, bu h, q = q P h. Traditional mixed-finite elements require the inf-sup condition to guarantee the existence of discrete solutions. As 2.6 provides a compatibility between the discrete velocity and discrete pressure spaces, the linear system of equations 2.7 always has a unique solution, independent of the inf-sup condition. 4

5 Proposition 2.. There is a unique solution in the discrete linear system 2.7. Proof. As 2.7 is a square system, the uniqueness implies existence. Let u h, p h be a solution for the homogeneous equations 2.7. Let v = u h and q = p h in 2.7. We have au h, u h + bu h, p h =, bu h, p h =. Subtracting the second equation from the first one, we get The first equation of 2.7 is now 2.8 au h, u h = u h =. bv, p h =, forall v V h. By 2.6, we have some w h V h such that p h = div w h. Let v = w h in 2.8. It is then div w h 2 L 2 =. So p h = div w h =. Further, by the second equation in 2.7 and the definition of P h in 2.6, we conclude that 2.9 bu h, q = bu h, div u h = div u h 2 L 2 Ω 2 = and that div u h =, i.e. u h is divergence-free. In this case, we call the mixed finite element a divergence-free element. It is apparent that the discrete velocity solution is divergence-free if and only if the discrete pressure finite element space is the divergence of the discrete velocity finite element space, i.e., 2.6. In fact, it is trivial to show [5, 6, 26], in the next theorem, that u h is the unique a, orthogonal projection from the divergence-free space Z to its subspace Z h, defined by That is, Z := { v H Ω 2 div v = }, Z h := {v V h div v = }. u h Z h, au u h, v h = v h Z h. We note that the definition of P h is abstract, which may not be good enough for computation the basis of functions of P h is unknown. But as the pressure space is defined implicitly by the velocity space, the unknown pressure solution can be implicitly defined by a function in the velocity space, by an iterative method, cf. [27]. Indeed, such a computation saves half of coding work on the pressure finite element and one-third unknowns in the resulting linear system of equations. But we do give another, traditional definition of the pressure space P h below. By 2.6, P h may not be the full 5

6 space of discontinuous P 2 polynomials, but a proper subspace if singular vertices are present. The next definition describes P h precisely. 2.2 P h = {p h L 2 Ω p h K P 2 K T h, i i= i v h Ki s = at a singular vertex s }, where s is one of the four types of vertexes depicted in Figure 2., and K i are the i =, 2, 3, 4 triangles around the singular vertex s. Lemma 2.. The dimension of V K in 2.3 is 26. The 26 degrees of freedom are listed in 2.4. Proof. dim P4 2 = 3. Adding the four constraints of P 3 coefficients of the divergence to the 26 degrees of freedom, we have a square system of linear equations. The existence of the solution is implied by the uniqueness, which will be proved next. The proof is done in two steps, on the reference triangle ˆK and on the general triangle K. Let the reference triangle be 2.3 ˆK = {ˆx, ŷ, ˆx + ŷ }. With all 26 dof in 2.4 of ˆv h be zero, ˆv h = on the boundary of ˆK, as each component of ˆv êi is a degree 4 polynomial with 5 zeros. Thus c + c ˆv h = ˆxŷ ˆx ŷ 2ˆx + c 3 ŷ 2.4, c 4 + c 5ˆx + c 6 ŷ for some constants c,... c 6. We show these constants are all zero. The ˆx 3 and ŷ 3 coefficients of divˆv h are c 5 and c 3, respectively. Thus c 3 = c 5 =. The ˆx 2 y coefficient of divˆv h is 3c 2 2c 5 2c 6 = 3c 2 2c 6 =. Similarly, by checking the ˆxŷ 2 coefficient of divˆv h, we have 2c 2 2c 3 3c 6 = 2c 2 3c 6 =. Together, we get c 2 = c 6 =. Thus c ˆv h = ˆxŷ ˆx ŷ. c 4 Because the bubble function ˆxŷ ˆx ŷ is positive inside ˆK, by the th moment of ˆv h on ˆK in 2.4, c = c 4 =. Thus ˆv h =. Now, we show the uniqueness on a general triangle K T h. Let F K be an affine mapping from ˆK to K, cf. Figure 2.2, such that x F K ˆx, ŷ = x + Bˆx = + ˆx 2.5 x y x x x 2. ŷ Now, if the 26 dof s of function v h have value, then v h is identically zero on the three edges of K: p v h = λ λ 2 λ 3, 6 p 2

7 ˆK : ˆx 2 ˆx ˆx F K = Bˆx + x x K : x 2 x Figure 2.2. An affine mapping F K from the reference triangle ˆK to a general triangle K. where λ i are three area-coordinates on K, and p i are two P polynomials. We define a Piola transform by 2.6 ˆv h ˆx = B v h F K ˆx, where B is defined in 2.5. Because v h is identically zero on the boundary of K, so is ˆv h. On the other side, if v h on the boundary of K, we cannot use the Piola transformation as it would destroy the tangential continuity of H functions. The ˆv h defined in 2.6 can also be expressed as 2.4. Since the Piola transform preserves the divergence, div v h = traceb T ˆ Bˆvh ˆx T = trace ˆ ˆvh ˆx T = divˆvh, divˆv h is also a P 2 function. Further, as B is invertible, two linear combinations of 2 zero -moment component of v h have also zero -moment. Thus, by the analysis on ˆv h above, ˆv h =. So is v h = Bˆv h F K x =. 3. Stability and convergence In this section, we will prove the on-to mapping property of the divergence operator, from the discrete velocity space to the finite element pressure space. Consequently we prove the inf-sup stability condition and the optimal order of convergence for the finite element solution. Remark 3.. The inf-sup condition 3., i,e, Theorem 3., can be proved as a corollary of Soctt-Vogelius Theorem 5. [8]. But we give an independent proof. The difference between the proof here and the Scott-Vogelius proof is in the construction of u h in next lemma, Lemma 3.. We use bubbleenriched P 3 polynomials equivalently P 4 polynomials to construct u h while Scott and Vogelius used only P 3 polynomials, cf. [2, Lemma 2.3] and [8, Lemma 4.]. Thus the Scott-Vogelius result requires the grid size h sufficiently small, [8, Remark 5., Lemma 5., and 5.6]. But the theory here does not have any restriction on h. 7

8 Lemma 3.. There is a u h V h, supported on two triangles K, K, cf. Figure 3., such that div u h has a nodal value at x on the K side, and nodal value at the rest P 2 Lagrange nodes on the two triangles, as long as the two edges x x and x x,k do not fall into a same line. ˆK ˆK x 2 = x,k K x K div u h = x = x 2,K Figure 3.. C -P 2 Lagrange nodes on two neighboring triangles and the reference mapping. x,k Proof. On the reference triangle ˆK in Figure 3., we find all such vectors û, which vanish on the boundary of ˆK ˆK and whose divergence is the P 2 polynomial having value at vertex ˆx only. divû ˆx, ŷ = 2ˆx 2 ˆx, on ˆK. Note that there are precisely three divergence-zero vectors. They are the curl of functions ˆx 3 ŷ 2, ˆx 2 ŷ 2 and ˆx 2 ŷ 3. So, on ˆK, 3. û ˆx, ŷ = ˆxŷ 2ˆx +c 2 2ˆx 2ŷ 2ˆx 2 + c 3ˆxŷ 3ˆxŷ + c 3 2ŷ 2. On the other reference triangle ˆK in Figure 3., we need divû =. So the vector must be a linear combination of the curls of, on ˆK, ˆx 2 ŷ 3, ˆx 2 ŷ 2 and ˆx 3 ŷ ˆx ŷ û ˆx, ŷ = ˆx ŷ c 4 2 ŷ 2 +c 5 2 ˆx 2 ŷ 8 + c 6 2 ˆx 2 3 ˆx ŷ.

9 We map these two vector functions to triangles K and K, preserving the divergence. Because û = on the four outside edges of triangles ˆK and ˆK, after the Piola transformation, the piecewise P 4 vector function remains zero on the four outside edges of triangles K and K. Then we only match the interface values of two P 4 vector functions at the common edge x x 2. Each P 4 function has 3 internal Lagrange nodal-values on an edge. We end up with 6 equations for the matching on x x 2. Though we have 6 degrees of freedom in 3. and 3.2, the system does not have a unique solution, as the curl of the C -P 5 Argyris normal derivative basis at the mid-point of x x 2 is a solution of the homogeneous system. We have either no solution, if x x and x x,k are colinear, or infinitely many solutions, if the two lines are different. We will show the detail next. The reference mapping from ˆK to K is defined in 2.5. On K, u is defined by the Piola transformation, 3.3 u x = Bû F K x. The reference mapping from ˆK to K is x y = F K ˆx, ŷ = x,k + B ˆx, ŷ where B = x,k x,k x,k x 2,K. In order to keep the divergence, similar to the Piola transformation 3.3, we let, on K, 3.4 u x, y = B û, F K x. When restricted on the common edge x x 2, i.e., ŷ = ˆx, we have 2ˆx B 2 c + 2ˆxc 2 + 3ˆx ˆx 2 c 3 2ˆx + 3 ˆx + ˆx 2 c ˆxc ˆx 2ˆx 2 c 3 = B 2ˆx 2 c 4 + 2ˆxc 5 + 3ˆx ˆx 2 c 6 3 ˆx + ˆx 2 c ˆxc ˆx 2ˆx 2. c 6 Matching coefficients of, ˆx and ˆx 2 of the two components, we have the following 6 6 system of equations, in block matrix form, A A A c c 2 c 3 c 4 c 5 c 6 = 2,

10 where the common matrix A is defined in A = B B = x x x x 2 x,k x,k x,k x 2,K = x x x x 2 x,k x 2 x,k x = x x x x 2 x x x x 2 + x,k x = + B x,k x B x,k x = + z z. z 2 + z 2 x,k x z Here = B z x,k x. By adding the first block and the third block to 2 the second block in the linear system, we have a simplified linear system 3.6 2z 2z z 2 2 2z z 2 + 2z 2z 2 2z z 3 5z z 2 2 5z 2 c c 2 c 3 c 4 c 5 c 6 =. There are two cases, z 2 = or z 2. z 2 = if and only if x,k x = cx x i.e. x,k is on the straight line x x. If z 2 =, by the fourth equation in 3.6, there is no solution. When z 2, we let c 6 = or a new constant. By the first equation, if z, c 5 =. But if z =, i.e., x,k is on the straight line x x 2, we have another degree of freedom and we also let it be zero, i.e., c 5 =. By the fourth equation in 3.6, c 4 = /2z 2. By this time, the system 3.6

11 is reduced to R +R 2 R 2 +R 3, 3/2R 2 +R 4 R 4 +R 3,2R 4 +R 2,2R 4 +R Thus, we find a solution z /z z /2z z 2 /2z z + z 2 /z z /2z z 2 /2z z + z 2 /z 2 3z 2z 2 /2z 2 3z + 2z 2 /2z 2 2 3z + 3z 2 /z 2 2 2z + 3z 2 /z 2. 3z + 2z 2 /2z 2 c 3 = 3z 2z 2 + c 2 = 3z 2z c = 2z 2z Lemma 3.2. There is a u h V h, supported on two triangles K, K, cf. Figure 3.2, such that div u h has a nodal value at x on both K and K, and nodal value at the rest P 2 Lagrange nodes on the two triangles when the two edges x x and x x,k fall into a same line. Proof. The proof repeats that for Lemma 3.. We still give the details. On the reference triangle ˆK in Figure 3.2, we find all such vectors û h, which vanish on the boundary of ˆK ˆK and whose divergence is the P 2 polynomial having value at vertex ˆx only. divû h ˆx, ŷ = 2ˆx 2 ˆx, on ˆK. Note that there are precisely three divergence-zero vectors. They are the curl of functions ˆx 3 ŷ 2, ˆx 2 ŷ 2 and ˆx 2 ŷ 3.

12 ˆK ˆK x 2 = x,k F K, F K K K x x,k x = x 2,K div u h = Figure 3.2. C -P 2 Lagrange nodes on two neighboring triangles and the reference mapping. So, on ˆK, 3.7 û h ˆx, ŷ = ˆxŷ 2ˆx +c 2 2ˆx 2ŷ 2ˆx 2 + c 3ˆxŷ 3ˆxŷ + c 3 2ŷ 2. On the other reference triangle ˆK in Figure 3.2, similarly, we have 3.8 2ŷ û h ˆx, ŷ = ˆx ŷ 2 ˆx +c 5 2 ŷ 3 ˆx ŷ + c 4 2 ŷ 2 + c 6 2 ˆx 2 3 ˆx ŷ. We map these two vector functions to triangles K and K, preserving the divergence, noting that they vanish on the outside of two edges of the triangles where they are defined. Then we match the interface values of two P 4 vector functions at the common edge x x 2. The reference mapping from ˆK to K is defined in 2.5. In order to keep the divergence, we define the two Piola transformations in 3.3 and 3.4. When restricted on the common edge x x 2, i.e., ŷ = ˆx, we have 2ˆx B 2 c + 2ˆxc 2 + 3ˆx ˆx 2 c 3 2ˆx + 3 ˆx + ˆx 2 c ˆxc ˆx 2ˆx 2 c 3 = B 2ˆx 2 c 4 + 2ˆxc 5 + 3ˆx ˆx 2 c 6 2ˆx + 3 ˆx + ˆx 2 c ˆxc ˆx 2ˆx 2. c 6 2

13 Matching coefficients of, ˆx and ˆx 2 of the two components, we have the following 6 6 system of equations, in block matrix form, c A c 2 A A c c 4 = A 2 2, c A where the common matrix A is defined in 3.5. As x,k is on the straight line x x, z 2 = in 3.5, and z unless the two triangles degenerate to a line segment. + z z A =. By the row operations of 3.6, it follows that 2z 2z z 2 + 2z z 3 5z c 6 c c 2 c 3 c 4 c 5 c 6 z = z. From the first equation, we choose c 6 = and c 5 = /2. Then adding the second equation to the third equation, we can let c = c 3 = c 4 = in the new third equation to satisfy it. By the second equation, c 2 = c 5 = /2. Thus, we find a solution u h which also satisfies, cf. Figure 3.2, 3.9 div u h dx = 2ˆx 2 ˆxdˆx =, K ˆK 3. div u h dx = ŷ 2ŷdˆx =. K ˆK Theorem 3.. For any q h P h 2.6, there is a v h V h 2.5, such that 3. div v h = q h and v h H C q h L 2, where C is independent of h, but dependent on the first level grid T h. Proof. As q h P h L 2 Ω, there is an H vector v H Ω2 such that, by [4, 6], div v = q h and v H C q h L 2. Let v = I h v be the Scott-Zhang [2] interpolation of v, by the 26 degrees of freedom in 2.4 so that the edge flux is preserved. Then v H 3

14 C v H, and K div v dx = = K K v nds = v nds K div vdx = q h dx, on every triangle K T h. Let q = q h div v P h. There are two types of vertices in triangulation T h, singular points and non-singular points. If s is an internal singular point, shown in Figure 3.3, the other end point of an edge having s as an end point points s, s 2, s 3, s 4 in Figure 3.3 cannot be a singular point, as the sum of two angles at s is already π. At s, the P h function q has three degrees of freedom, not four, i.e., K 3.2 q K q K2 + q K3 q K4 s =, cf. Figure 3.3. s s 2 K K 2 s K 4 K 3 s 4 s 3 Figure 3.3. An internal singular point s and its four neighboring triangles. Let v,2 P 3 K P 3 K 2 H K K 2 be such that v 2 s = 2 vs 2 = 2 vs 3 =, ss2 v 2 =, where ss2 denotes the directional derivative along the direction from s to s 2. That is, on K, v,2 = cλ ss λ 2 s s 2, c = ss2 λ ss λ s s 2 s 2, where λ ss is the linear function assuming on edge ss and at the opposite vertex s 2. On K 2, v 2 is defined symmetrically. Let v 2 = [q K sv 2 ]t ss2, 4

15 where t ss2 is the unit tangent vector along the direction from s to s 2, cf. Figure 3.3. It follows that div v 2 K s = q K s [t ss2 x v 2 s + t ss2 2 x2 v 2 s] = q K s ss2 v 2 s = q K s. But, on K 2, div v 2 s = q K s, not matching q K2 s yet. At the rest nodes and the node s of other triangles, div v 2 Kj s i =. Also, due to the tangent vector in the definition, via integration by parts, we have div v 2 dx = v 2 nds = ds =. K K K The integral is also zero on K 2. Similarly, we define Together, we let v 23 = q K2 s q K sv 23 t ss3, v 34 = q K3 s q K2 s + q K sv 34 t ss4. v s = v 2 + v 23 + v 34. By the construction, div v s s = q s, on K, K 2 and K 3. By the constraint 3.2, on K 4, div v s s = q K3 s q K2 s + q K s = q K4 s. So the divergence of the constructed v s matches the four values of q at the singular point s. By the scaling argument, we have v s H Ω Ch max i 4 q Ki s C q L 2 4 i= K i. In the same way, we can define a discrete velocity at each singular point, including the boundary singular points depicted in Figure 2.. Summing these velocity functions, we name it v 2 such that { q s at all singular vertices on all triangles, div v 2 s = at rest vertices on all triangles, div v 2 dx = on all triangles, K v 2 H Ω C q L 2 Ω. Let q 2, = q div v 2 P h. The rest vertices are non-singular. But there is a special type of nonsingular vertex, shown in Figure 3.4, that the triangle K and both its neighboring triangles at vertex x, K and K, form a straight line passing through x. The matching at x on K must be done before the matching on the rest triangles at x can be done by the above method for treating the singular vertex. But we give another construction in Lemma 3.2. Let x be a non-singular vertex on the middle triangle K, cf. Figure

16 x K K x K x 4 x 2 x 3 Figure 3.4. A non-singular point x in K with two straight-line neighboring triangles, K and K. By lemma 3.2, 3.9 and 3., there is a v x,k,k V h H K K 2, such that div v x,k,k K x = div v x,k,k K x = q 2, K x, div v x,k,k Ki x j = at rest vertices x j on triangles K i = K, K, K i div v x,k,k dx = on all triangles K i = K, K, v x,k,k H Ω Ch q2, K x C q2, L 2 K K. There is likely no such half-singular vertex. But if there are, we sum such velocity functions v x,k,k and name it v 2,. Let q 2 = q 2, div v 2, P h. q 2 keeps all properties of q 2, except having different nodal values at half-singular vertex that q 2 K x = and q 2 K x = q 2, K x q 2, K x. For each x of the rest vertices on one of its associated triangle K it must have at least one neighboring triangle K not forming a straight line with K, by Lemma 3., there is a vector v x,k V h, supported on K and another neighboring triangle K, such that 3.3 div v x,k K x = q 2 K x, div v x,k K x = at rest vertices x and triangles K, div v x,k dx = on all triangles K, K v x,k H Ω Ch q 2 K x C q 2 L 2 K K. 6

17 Here 3.3 holds because, in the construction of v x,k in Lemma 3., we have v x,k C K K 2 and div v x,k K = so that div v x,k dx = div v x,k dx div v x,k dx K K K K = v x,k nds dx K K K = ds =. K K Once more, summing all these v x,k, we get a vector function v 3 P h, such that div v 3 matches all non-zero vertex values of q 2, and does not destroy the properties by the constructions v and v 2. Let q 3 = q 2 div v 3 P h. On each triangle, q 3 is P 2 polynomial vanishing at three vertices and having mean value K q 3dx =. On a triangle K, the dimension of q 3 functions is 2, not 3. We construct a P 3 vector bubble on each triangle so that its divergence matches two of the mid-edge values, q 3 m and q 3 m 2, of q 3 on the triangle. Let v K = B K q 3 m t + q 3 m 2 t 2 where B K = λ λ 2 λ 3, A = B K m B K m 2, t t 2 = A T, and λ i is the linear function vanishing on edge e i and assuming value on the opposite vertex of K. Note that, the matrix A is invertible as the triangle K is non-singular. By this construction, we have div v K m = x B K m q 3 m t + q 3 m 2 t 2 + x2 B K m q 3 m t + q 3 m 2 t 2 2 = q 3 m B K m t + q 3 m 2 B K m t 2 = q 3 m + q 3 m 2 = q 3 m. Also div v K m 2 = q 3 m 2. As B K has two directional derivatives at the vertices, div v K x i = at all three vertices. Also as B K K =, K div v K = K v K n =. That is, the P 2 polynomial q 3 div v K, on K, has 5 zero values at 5 Lagrange nodes and zero mean value. Thus q 3 = div v K on K. Summing all such v K of all triangles K T h, we let it be v 4 that div v 4 = q 3, v 4 H Ω Ch q 3 L Ω C q 3 L 2 Ω. Combining the four constructed vectors, we let v h = v + v 2 + v 2, + v 3 + v 4. 7

18 Then div v h = p h, and, due to the finite over-lapping, v h H v H + v 2 H + v 2, H + v 3 H + v 4 H v H + v 2 H + v 2, H + v 3 H + C q 3 L 2 v H + v 2 H + v 2, H + C v 3 H + C q 2 L 2 C v H + C q L 2 C q h L 2. Because of the homogeneous boundary condition, the divergence operator may not have a bounded right inverse in an high order Sobolev norm, even if the solution p is smooth, [4, Theorem 3.]. So we assume, by [4, Theorem 3.], for the solution p of 2., there is a w H Ω2 H 4 Ω 2 such that 3.4 div w = p and w H 4 C p H 3. Theorem 3.2. Let the grids be defined by the multigrid refinement of an initial grid T h. The finite element solution u h, p h of 2.7 is quasi-optimal in approximating the exact solution u, p of the Stokes equation 2., assuming 3.4 holds, u u h H + p p h L 2 C inf v h V h,q h P h u v h H + p q h L 2 Ch 3 u H 4 + p H 3. Proof. By 3., the following inf-sup condition holds, inf q h P h div v h, q h sup C v h V h v h H q h L 2 with C independent of the grid size h. By the standard theory on saddlepoint approximation [5, 6], the quasi-optimality inequality holds. By the Scott-Zhang interpolation operator [2], we have inf v h V h u v h H C u I h u H Ch 3 u H 4, inf q h P h p q h L 2 C div w div I h w L 2 Ch 3 w H 4 Ch 3 p H Numerical tests We solve the Stokes 2. on the unit square Ω =, 2, where the exact solution is 4. u = curl g, p = g, where g = 2 8 x 2 x 2 y 2 y 2. The first grid is the northwest-southeast cut of the domain, shown in Figure 4.. Then the standard multigrid refinement is applied to generate higher levels of grids, shown in Figure 4.. For such grids, there are only two 8

19 Figure 4.. The level, 2 and 3 uniform grids. Table 4.. The errors, e u = u I u h and e p = p I p h, and the order of convergence, by the P 3 + B 4 /P 2 finite element 2.5 and 2.6, for the problem 4.. e u L 2 h n e u H h n e p L 2 h n dim V h singular-points, the northeast corner and the southwest corner, i.e.,, and,. We first apply the newly proposed P + 3 -P 2 mixed finite element method 2.5. The output data are listed in Table 4.. As proved in Theorem 3.2, the finite element solution converges at the optimal order. Table 4.2. The errors, e u = u I u h and e p = p I p h, and the order of convergence, by the P 4 /P 3 Scott-Vogelius finite element, for the problem 4.. e u L 2 h n e u H h n e p L 2 h n dim V h In the second numerical test, we compute the solution 4. again by the Scott-Vogelius P 4 -P 3 element, the lowest order stable element of [8]. The error and the order of convergence are listed in Table 4.2. The optimal order of convergence is obtained there. In the third numerical test, we use an unstable mixed finite, the continuous P 3 velocity with discontinuous P 2 pressure subset, P h = divc -P 3, to solve the Stokes equation 4.. The resulting linear system of equations is solved by the iterated-penalty method, cf. [26, 27]. The error and the 9

20 Table 4.3. The errors, e u = u I u h and e p = p I p h, and the order of convergence, by the P 3 -P 2 finite element, for the problem 4.. e u L 2 h n e u H h n e p L 2 h n dim V h order of convergence are listed in Table 4.3. In this case, the pressure converges two orders below the optimal order, to the true solution. The finite element velocity is one order sub-optimal. But as the inf-sup condition fails, we do not have a theory to cover the above observation, i.e., the sub-optimal convergence is not proved. Acknowledgment We thank an anonymous referee for pointing out Remark 3., and Figure 3.4 which was not discussed in an earlier version of the paper. This research is partially supported by the NSFC project References [] D. N. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements, in Advances in Computer Methods for Partial Differential Equations VII, ed. R. Vichnevetsky and R.S. Steplemen, 992. [2] D. N. Arnold and R. Winther, Mixed finite elements for elasticity, Numer. Math no. 3, [3] D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comp , no. 263, [4] D. Arnold, L.R. Scott and M. Vogelius, Regular inversion of the divergence operator with Dirichlet conditions on a polygon, Ann. Sc. Norm. Super Pisa, C. Sci., IV Ser., 5988, [5] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 994. [6] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer, 99. [7] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 978. [8] R. Falk and M. Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM J. Numer. Anal. 5 23, no. 2, [9] J. Guzman and M. Neilan, Conforming and divergence free Stokes elements on general triangular meshes, Math. Comp , no. 285, [] J. Guzman and M. Neilan, Conforming and divergence-free Stokes elements in three dimensions, IMA J. Numer. Anal , no. 4, [] J. Hu and S. Zhang, A family of conforming mixed finite elements for linear elasticity on triangle grids, arxiv: [math.na]. 2

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