Analysis of Piecewise Linear Approximations to the Generalized Stokes Problem in the Velocity–Stress–Pressure Formulation Suh-Yuh Yang
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Journal of Computational and Applied Mathematics 147 (2002) 53–73 www.elsevier.com/locate/cam
Analysis of piecewise linear approximations to the generalized Stokes problem in the velocity–stress–pressure formulation Suh-Yuh Yang
Department of Mathematics, National Central University, Chung-Li 32054, Taiwan Received 26 March 2001; received in revised form 10 December 2001
Abstract
In this paper we study continuous piecewise linear polynomial approximations to the generalized Stokes equations in the velocity–stress–pressure ÿrst-order system formulation by using a cell vertex ÿnite volume= least-squares scheme.This method is composed of a direct cell vertex ÿnite volume discretization step and an algebraic least-squares step, where the least-squares procedure is applied after the discretization process is accomplished.This combined approach has the advantages of both ÿnite volume and least-squares approaches. An error estimate in the H 1 product norm for continuous piecewise linear approximating functions is derived.It is shown that, with respect to the order of approximation for H 2-regular exact solutions, the method exhibits an optimal rate of convergence in the H 1 norm for all unknowns, velocity, stress, and pressure. c 2002 Elsevier Science B.V. All rights reserved.
MSC: 65N15; 65N30; 76M10
Keywords: Generalized Stokes equations; Velocity–stress–pressure formulation; Elasticity equations; Finite volume methods; Least squares
1. Introduction
Continuous piecewise linear polynomials may be the most natural and simple approximating func- tions in the numerical schemes, especially for large-scale computations, for solving partial di This work was supported in part by the National Science Council of Taiwan, under grants NSC 89-2115-M-008-030 and NSC 90-2115-M-008-008. E-mail address: [email protected] (S.-Y. Yang). 0377-0427/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S 0377-0427(02)00392-8 54 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 with the homogeneous Dirichlet velocity boundary condition in the velocity–stress–pressure ÿrst-order system formulation by using a cell vertex ÿnite volume=least-squares numerical scheme.It is shown that, with respect to the order of approximation for H 2-regular exact solutions, this combined scheme exhibits an optimal rate of convergence in the H 1 norm for all unknowns (velocity, stress, and pressure). This study is motivated by some signiÿcant observations from the numerical experiments of least-squares ÿnite element methods [8].Over the past decade, least-squares ÿnite element meth- ods have become increasingly popular for the approximate solution of ÿrst-order systems of partial di 2. Problem formulation Let be a bounded, open, and connected domain in Rd;d= 2 or 3, with Lipschitz boundary T 2 d @.Let f =(f1;:::;fd) ∈ [L ()] be a given vector function representing the density of body force.The stationary incompressible Navier–Stokes equations supplemented with the homogeneous Dirichlet velocity boundary condition can be posed as [14,15] − Ou +(u ·∇)u + ∇p = f in ; ∇·u =0 in ; (2.1) u = 0 on @; where the symbols , ∇, and ∇· stand for the Laplacian, gradient, and divergence operators, re- T spectively; u =(u1;:::;ud) is the velocity; p is the pressure satisfying the zero mean condition p d =0;0¡ 6 1 is the viscosity constant.All of variables are assumed to be nondimension- alized. According to the theory of Brezzi–Rappaz–Raviart [5], the formulation and analysis of discretiza- tion methods for the Stokes problem are critical for the understanding of similar methods for the Navier–Stokes problem.Thus, we restrict our attention to the Stokes Iows.Neglecting the nonlinear term in the above system, we have − Ou + ∇p = f in ; ∇·u =0 in ; (2.2) u = 0 on @: Further, for the sake of generality, we consider the following generalized Stokes equations supple- mented with the homogeneous Dirichlet velocity boundary condition − Ou + ∇p = f in ; ∇·u + p = g in ; (2.3) u = 0 on @; 56 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 where g ∈ H 1() is a given scalar function with g d =0; is a ÿxed nonnegative constant bounded uniformly in .When =0;g= 0 we have the Stokes problem, and the case of ¿0 arises from the linear elasticity problem.See Remark 2.1below for the details. When d = 2, we deÿne the curl operator “∇×” for smooth two-component vector function v = T (v1;v2) by ∇×v = @1v2 − @2v1: T When d = 3, we deÿne the curl of a smooth three-component vector function w =(w1;w2;w3) by T ∇×w =(@2w3 − @3w2;@3w1 − @1w3;@1w2 − @2w1) : Now, introducing the auxiliary d × d new independent variables T U = ∇u =(∇u1;:::;∇ud) called stresses here, we can transform (2.3) into the following so-called velocity–stress–pressure ÿrst-order system formulation − (∇·U)T + ∇p = f in ; U −∇uT = 0 in ; (2.4) ∇·u + p = g in ; u = 0 on @: Note that the deÿnition of U, the “continuity” equation ∇·u + p = g in , and the Dirichlet boundary condition u = 0 on @ imply the respective properties ∇×U = 0 in ; trU + p = g in ; (2.5) n × U = 0 on @; where the trace operator “tr” is deÿned as trU = U11 + ···+ Udd.Thus, an equivalent extended system for (2.4) is given by − (∇·U)T + ∇p = f in ; U −∇uT = 0 in ; ∇·u + p = g in ; ∇trU + ∇p = ∇g in ; (2.6) ∇×U = 0 in ; u = 0 on @; n × U = 0 on @: Refer to Table 1 for the numbers of unknowns, equations, and boundary conditions of problem (2.6) in di Table 1 Generalized Stokes system (2.6) Dim. No.of unknowns No.of equations No.of boundary conditions d2 + d +1 3d2 − d +1 2d2 − 2d d =2 7 11 4 d = 3 13 25 12 Remark 2.1. An important aspect of the pressure-perturbed form of the generalized Stokes problem (2.3) is that it allows our results to apply to the Dirichlet problem for the linear elasticity equations. In particular; consider the following Dirichlet problem −Ou − ( + )∇(∇·u)=f in ; u = 0 on @; where u now represents displacements and ; are the positive LamÃe constants.This system can be recast in form (2.3) by introducing the artiÿcial pressure variable p = − (∇·u) 2 by rescaling f; and by letting g =0;=2=; and = =(2( + )). We use the standard notation and deÿnition for the Sobolev spaces H m(), m ¿ 0, with inner 2 0 2 products (·; ·)m; and norms ·m;.As usual, L ()=H (), and L0() denotes the subspace of L2() that consists of square integrable functions with zero mean.Let [ H m()]d denote the corresponding product spaces, and the inner products and norms will be still denoted by (·; ·)m; and ·m;, respectively, when there is no chance of confusion.We will be interested in the following three function spaces with respect to the unknown functions, velocity u, stress U, and pressure p V = {v : v ∈ [H 1()]d and v = 0 on @}; (2.7) S = {V : V ∈ [H 1()]d2 and n × V = 0 on @}; (2.8) 1 Q = {q: q ∈ H () and (q; 1)0; =0}: (2.9) The existence, uniqueness and smoothness of the weak solution (u;p) to problem (2.3) are 1 d 2 2 d well-known.Problem (2.3)has a unique weak solution ( u;p) ∈ [H0 ()] ×L0() for any f ∈ [L ()] and g ∈ H 1() (see, e.g., [4,14,15]). Moreover, if the boundary @ of the domain is C1;1, then the following H 2 regularity result holds u2; + p1; 6 C(f0; + g1;): (2.10) When the domain is a convex polyhedron, a priori estimate (2.10) is still valid when ¿0.For the case = 0, one has only a weaker estimate (cf.[6]).Throughout this paper, in any estimate or inequality, the quantity C with or without subscripts will denote a generic positive constant always 58 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 independent of and the mesh parameter h, which will be introduced later, and need not be the same constant in di T 2 2 T 2 K((v; V;q)) ≡− (∇·V) + ∇q0; + V −∇v 0; 2 2 2 2 2 2 + ∇ · v + q0; + ∇trV + ∇q0; + ∇ × V0;: Theorem 2.1. Assume that the domain is a bounded convex polyhedron or has C1;1 boundary @ and that regularity estimate (2.10) holds. Then there exists a positive constant C independent of such that for any v ∈ V; V ∈ S; and q ∈ Q; we have 1 ( 2v2 + 2V2 + q2 ) 6 K((v; V;q)) C 1; 1; 1; 2 2 2 2 2 6 C( v1; + V1; + q1;): (2.11) Throughout the remainder of this paper, we will always assume that the conditions in Theorem 2.1 hold such that we have the a priori estimate (2.11). 3. The cell vertexÿnite volume scheme For ÿnite volume approximation, there are generally a pair of discretizations of the domain ,a primal partition and a dual partition.However, these two discretizations of are the same in the cell vertex ÿnite volume computations.Let {Th} be a family of regular triangulations of the domain (cf.[4,13]), where h Th = {k : k =1;:::;T(h)}; h h h=max{diam(k ): k ∈ Th} denotes the grid size, and T(h) denotes the number of triangles (cells). h h h h h Let k denote the area (volume) of the cell k , i.e., k =|k |.Let P1(k ) denote the space of linear h polynomials deÿned over k .Deÿne the following three continuous piecewise linear approximating function spaces with respect to the triangulation Th, h d Vh = {vh ∈ V: vh| h ∈ [P1( )] for k =1;:::;T(h)}; (3.1) k k h d2 Sh = {Vh ∈ S: Vh| h ∈ [P1( )] for k =1;:::;T(h)}; (3.2) k k h Qh = {qh ∈ Q: qh| h ∈ P1( ) for k =1;:::;T(h)}: (3.3) k k Then the ÿnite-dimensional function spaces Vh, Sh, and Qh satisfy the following approximation 2 d 2 d2 2 properties: for any v ∈ V ∩ [H ()] , V ∈ S ∩ [H ()] , and q ∈ Q ∩ H (), there exist vh ∈ Vh, S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 59 Vh ∈ Sh, and qh ∈ Qh such that 1 v − vh1; 6 Ch v2;; (3.4) 1 V − Vh1; 6 Ch V2;; (3.5) 1 q − qh1; 6 Ch q2;; (3.6) where C is a positive constant independent of v, V, q, and h. We now present the cell vertex ÿnite volume=least-squares algorithm.The ÿrst step is to form an overdetermined linear system of equations Determine (uh; Uh;ph) ∈ Vh × Sh × Qh such that 1 T 1 (− (∇·Uh) + ∇ph)d = f d; h h h h k k k k T (Uh −∇uh )d = 0; h h k k (∇·uh + ph)d = g d; h h h h k k k k (∇trUh + ∇ph)d = ∇g d; h h h h k k k k (∇×Uh)d = 0 (3.7) h h k k for k =1;:::;T(h). The equations in (3.7) are simply formed by integrating the components of the di AX = R: (3.8) The second step is to solve the overdetermined problem by the following algebraic least-squares problem 60 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 Determine X˜ such that ˜ AX − R2 = min AX − R2 (3.9) X or equivalently, to solve the normal equations Determine X˜ such that ATAX˜ = ATR: (3.10) 4. Error analysis This section is devoted to the error analysis for the continuous piecewise linear approximate solution (uh; Uh;ph) obtained by solving problem (3.10). We will show that problem (3.10) is equivalent to a minimization problem on which the error analysis is based.To this end, for each ÿxed triangulation Th, we deÿne the Th-dependent nonlinear functional Fh : V × S × Q → R by 2 T(h) 1 T Fh((v; V;q); f;g)= (− (∇·V) + ∇q − f)d h h k=1 k k 2 T + (V −∇v − 0)d h h k k 2 + (∇·v + q − g)d h h k k 2 + (∇trV + ∇q −∇g)d h h k k 2 + (∇×V − 0)d : h h k k Note that if (u; U;p) ∈ V × S × Q is the exact solution of problem (2.6), then Fh((u; U;p); f;g)=0: ˜ Therefore, we deÿne an approximate solution (u˜h; Uh; p˜ h)to(u; U;p) as follows: ˜ Determine (u˜h; Uh; p˜ h) ∈ Vh × Sh × Qh such that ˜ Fh((u˜h; Uh; p˜ h); f;g) = min Fh((vh; Vh;qh)); f;g): (4.1) (vh;Vh;qh)∈Vh×Sh×Qh S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 61 ˜ Since Fh((u˜h; Uh; p˜ h)+!(vh; Vh;qh); f;g) is a nonnegative quadratic functional in ! ∈ R, for any given (vh; Vh;qh) ∈ Vh × Sh × Qh, we have d F ((u˜ ; U˜ ; p˜ )+!(v ; V ;q ); f;g)| = 0 (4.2) d! h h h h h h h !=0 which is equivalent to ˜ Determine (u˜h; Uh; p˜ h) ∈ Vh × Sh × Qh such that ˜ Bh((u˜h; Uh; p˜ h); (vh; Vh;qh)) = Lh((vh; Vh;qh)) (4.3) for all (vh; Vh;qh) ∈ Vh × Sh × Qh, where the bilinear form Bh(·; ·) and the linear form Lh(·) are respectively deÿned as follows: Bh((v; V;q); (w; W;r)) T(h) 1 1 = (− (∇·V)T + ∇q)d · (− (∇·W)T + ∇r)d h h h h k=1 k k k k + (V −∇vT)d · (W −∇wT)d h h h h k k k k + (∇·v + q)d × (∇·w + r)d h h h h k k k k + (∇trV + ∇q)d · (∇trW + ∇r)d h h h h k k k k + (∇×V)d · (∇×W)d ; h h h h k k k k T(h) 1 1 T Lh((v; V;q)) = (− (∇·V) + ∇q)d · f d h h h h k=1 k k k k + (∇·v + q)d × g d h h h h k k k k + (∇trV + ∇q)d · ∇g d h h h h k k k k 62 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 for all (v; V;q); (w; W;r) ∈ V × S × Q.The unique solvability for problem (4.3)will be proved in Theorem 4.1 later. Now, it is easily seen, by choosing a basis for Vh × Sh × Qh, that problem (4.3) is equivalent to T ˜ T problem (3.10). That is, they have the same linear system problem A AX =A R and (uh; Uh;ph)= ˜ (u˜h; Uh; p˜ h).Therefore, throughout the remaining part of this section, we shall derive a priori error estimates for the approximate solution (uh; Uh;ph) based on the variational formulation (4.3). The next several lemmas will be useful in the proof of our main result, Theorem 4.2. Using the fact that Vh, Sh, and Qh are continuous piecewise linear function spaces (all the second derivatives h of functions in them vanish on k ), we have the following coercivity estimate for the bilinear form Bh(·; ·), which plays an important role in the error analysis. Lemma 4.1. The bilinear form Bh(·; ·) satisÿes the following coercivity property: |Bh((v; V;q); (vh; Vh;qh))| T T ¿ |(− (∇·V) + ∇q; − (∇·Vh) + ∇qh)0; T T +( (V −∇v ); (Vh −∇vh ))0; +( (∇·v + q); (∇·vh + qh))0; +( (∇trV + ∇q); (∇trVh + ∇qh))0; +( (∇×V); (∇×Vh))0;| − Ch( vh1; + Vh1; + qh1;) T T ×(− (∇·V) + ∇q0; + (V −∇v )0; + (∇·v + q)0; + (∇trV + ∇q)0; + (∇×V)0;); (4.4) for all (v; V;q) ∈ V × S × Q and (vh; Vh;qh) ∈ Vh × Sh × Qh. Proof. The proof is simple but long.The basic idea is based on the mean value theorem.Given h h d 1 k ∈ Th; assume ’ : k ⊂ R → R is a C mapping.Then the mean value theorem asserts that h for any x; y ∈ k ; there exists z ∈ xy; z = tx +(1− t)y for some 0 ¡t¡1; such that ’(x) − ’(y)=∇’(z) · (x − y): Taking integration with respect to y; we have ’(x) − ’(y)dy = ∇’(z) · (x − y)dy h h k k S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 63 which implies ’(x) 1dy = ’(y)dy + ∇’(z) · (x − y)dy h h h k k k and hence 1 1 ’(x)= ’(y)d + ∇’(z) · (x − y)d : (4.5) h y h y h h k k k k 2 T T Now let ( =3d − d +1, ) =( 1;:::; () , and Z =(+1;:::;+() be two vector-valued mappings S 2 h 1 h deÿned over , where i|h ∈ L (k ) and +i|h ∈ C (k ) for all i =1;:::;( and k =1;:::;T(h).Then, using (4.5), we have k k T(h) ( (); Z)0; = j(x)+j(x)dx h k=1 j=1 k T(h) 1 ( = (x) + (y)d + ∇+ (z ) · (x − y)d d h j j y j j;k y x h h h k=1 k j=1 k k k T(h) 1 ( = (x)d + (y)d h j x j y h h k=1 k j=1 k k T(h) 1 ( + (x) ∇+ (z ) · (x − y)d d : h j j j;k y x h h k=1 k j=1 k k Now taking absolute values on the both sides of the above equation, we can conclude that T(h) 1 |(); Z) | 6 ) d · Z d 0; h x y h h k=1 k k k T(h) 1 ( + | (x)| |∇+ (z ) · (x − y)| d d ; (4.6) h j j j;k y x h h k=1 k j=1 k k h where zj;k = tj;k x +(1− tj;k )y ∈ k for some tj;k ∈ (0; 1), j =1;:::;(. We are now in the position to prove the lemma.First of all, we set ) = − (∇·V)T + ∇q; (V −∇vT); (∇·v + q); (∇trV + ∇q); (∇×V) 2 h ( for (v; V;q) ∈ V × S × Q.Then )|h ∈ [L (k )] for all k =1;:::;T(h).Similarly, we set k T T Z = − (∇·Vh) + ∇qh; (Vh −∇vh ); (∇·vh + qh); (∇trVh + ∇qh); (∇×Vh) 1 h ( for (vh; Vh;qh) ∈ Vh × Sh × Qh.Then Z| h ∈ [C ( )] for all k =1;:::;T(h).Recall that vh, Vh, k k h and qh are continuous piecewise linear polynomials on , and diam(k ) 6 h.Thus we have the 64 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 following estimates: for x; y ∈ h, and z = t x +(1− t )y for some t ∈ (0; 1), k j;k j;k j;k j;k T I1j(x) ≡ |∇((− (∇·Vh) + ∇qh)j)(zj;k ) · (x − y)| dy h k T 6 |∇((− (∇·Vh) + ∇qh)j)(zj;k )(x − y)| dy h k T 6 h |∇((− (∇·Vh) + ∇qh)j)(zj;k )| dy h k =0; (4.7) T I2j(x) ≡ |∇( (Vh −∇vh )j)(zj;k ) · (x − y)| dy h k T 6 |∇( (Vh −∇vh )j)(zj;k )(x − y)| dy h k T 6 h |∇( (Vh −∇vh )j)(zj;k )| dy h k h 6 h k ∇Vh0;h ; (4.8) k I3(x) ≡ |∇( (∇·vh + qh))(zj;k ) · (x − y)| dy h k 6 h |∇( (∇·vh + qh))(zj;k )| dy h k h 6 h k ∇qh0;h ; (4.9) k I4j(x) ≡ |∇( (∇trVh + ∇qh)j)(zj;k ) · (x − y)| dy h k 6 |∇( (∇trVh + ∇qh)j)(zj;k )(x − y)| dy h k 6 h |∇( (∇trVh + ∇qh)j)(zj;k )| dy h k =0; (4.10) I5j(x) ≡ |∇( (∇×Vh)j)(zj;k ) · (x − y)| dy h k 6 |∇( (∇×Vh)j)(zj;k )(x − y)| dy h k S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 65 6 h |∇( (∇×Vh)j)(zj;k )| dy h k =0: (4.11) Substituting (4.7)–(4.11) into (4.6) then yields T T (− (∇·V) + ∇q; − (∇·Vh) + ∇qh)0; T T +( (V −∇v ); (Vh −∇vh ))0; +( (∇·v + q); (∇·vh + qh))0; +( (∇trV + ∇q); (∇trVh + ∇qh))0; +( (∇×V); (∇×Vh))0;| 6 |Bh((v; V;q); (vh; Vh;qh))| T(h) 1 d + |(− (∇·V)T + ∇q) (x)|I (x)d h j 1j x h k=1 k j=1 k d2 T + | (V −∇v )j(x)|I2j(x)dx h j=1 k + | (∇·v + q)(x)|I3(x)dx h k d + | (∇trV + ∇q)j(x)| I4j(x)dx h j=1 k 2d 2−3d + | (∇×V)j(x)|I5j(x)dx h j=1 k 6 |Bh((v; V;q); (vh; Vh;qh))| T(h) T + h (V −∇v ) h ∇Vh h 0;k 0;k k=1 + h (∇·v + q) h ∇qh h 0;k 0;k 6 |Bh((v; V;q); (vh; Vh;qh))| T(h) + Ch vh h + Vh h + qh h 1;k 1;k 1;k k=1 66 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 T T × − (∇·V) + ∇q h + (V −∇v ) h + (∇·v + q) h 0;k 0;k 0;k + (∇trV + ∇q) h + (∇×V) h 0;k 0;k 6 |Bh((v; V;q); (vh; Vh;qh))| + Ch vh1; + Vh1; + qh1; T T × − (∇·V) + ∇q0; + (V −∇v )0; + (∇·v + q)0; + ∇trV + ∇q)0; + (∇×V)0; : This completes the proof. Lemma 4.2. There exists an h0 ¿ 0 such that; for su4ciently small h¡h0; 2 2 2 2 2 Bh((vh; Vh;qh); (vh; Vh;qh)) ¿ C( vh1; + Vh1; + qh1;); (4.12) for all (vh; Vh;qh) ∈ Vh × Sh × Qh. Proof. Combining (4.4) with (2.11); we get for all (vh; Vh;qh) ∈ Vh × Sh × Qh; 2 2 2 2 2 Bh((vh; Vh;qh); (vh; Vh;qh)) ¿ C( vh1; + Vh1; + qh1;) 2 2 2 2 2 − Ch( vh1; + Vh1; + qh1;): Thus; for suHciently small h; we obtain (4.12). This completes the proof. The coercivity property (4.12) ensures the unique solvability of problem (4.3) or equivalent prob- lem (3.10), and also gives an estimate of the condition number of the matrix ATA in (3.10). Theorem 4.1. There exists an h0 ¿ 0 such that problem (4.3); or equivalent problem (3.10); has T a unique solution whenever h¡h0. In this case; the symmetric matrix A A appearing in (3.10) is positive deÿnite. Moreover; if the family {Th} of regular triangulations of the domain is quasi-uniform; then the condition number of ATA is O( −2h−2). Proof. For suHciently small h¡h0; by (4.12); we have seen the bilinear form Bh(·; ·) is coercive on the continuous piecewise linear function space Vh ×Sh ×Qh.The positive deÿniteness of the matrix ATA appearing in problem (3.10) follows immediately. Thus; the unique solvability of problem (4.3) is also ensured. Next, we give an estimate for the condition number of the linear system arising from problem (3.10). Recall that the condition number for a symmetric and positive deÿnite ‘ × ‘ matrix M is deÿned by max R(0) condition number of M ≡ max = ; min min R(0) S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 67 where max and min are the largest and smallest eigenvalues of M, respectively, and R(0)isthe Rayleigh quotient 0TM0 R(0) ≡ for all 0 =(1 ;:::;1 )T ∈ R‘;0∈ 0: 0T0 1 ‘ 1 m 1 n 1 k Let v ;:::;v , V ;:::;V , and q ;:::;q be bases for Vh, Sh, and Qh, respectively.Since the family {Th} of triangulations is regular [13], there exist positive constants 31, 32, 41, 42,T1, and T2 such that, for any vh ∈ Vh, Vh ∈ Sh, and qh ∈ Qh with m n k i i i vh = 1iv ; Vh = ÁiV ; and qh = +iq ; i=1 i=1 i=1 we have m m d 2 2 d 2 31h 1i 6 vh0; 6 32h 1i ; (4.13) i=1 i=1 n m d 2 2 d 2 41h Ái 6 Vh0; 6 42h Ái ; (4.14) i=1 i=1 k k d 2 2 d 2 T1h +i 6 qh0; 6 T2h +i : (4.15) i=1 i=1 S If, in addition, the corresponding regular family {Th} of triangulations of is quasi-uniform [13], i.e., there exists a positive constant C independent of h such that h h h 6 C diam(k ) for all k ∈ Th; Th ∈{Th} (4.16) then we have the following inverse estimates [13] −1 vh1; 6 Ch vh0;; (4.17) −1 Vh1; 6 Ch Vh0;; (4.18) −1 qh1; 6 Ch qh0;; (4.19) where C is a positive constant independent of h. m i n i k i For any vh = i=1 1iv ∈ Vh, Vh = i=1 ÁiV ∈ Sh, and qh = i=1 +iq ∈ Qh, by (2.11), (4.13), (4.14), and (4.15), we have 2 2 2 2 2 B((vh; Vh;qh); (vh; Vh;qh)) ¿ C( vh1; + Vh1; + qh1;) 2 2 2 2 2 ¿ C( vh0; + Vh0; + qh0;) m n k 2 d 2 2 2 ¿ C min{31;41; T1} h 1i + Ái + +i : i=1 i=1 i=1 68 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 On the other hand, it follows from (2.11), (4.17), (4.18), and (4.19) that 2 2 2 2 2 B((vh; Vh;qh); (vh; Vh;qh)) 6 C( vh1; + Vh1; + qh1;) −2 2 2 2 2 2 6 Ch ( vh0; + Vh0; + qh0;) m n k 2 d−2 2 2 2 6 C max{32;42; T2}(1 + )h 1i + Ái + +i : i=1 i=1 i=1 2 d−2 2 d Thus, max 6 C max{32;42; T2}(1+ )h and min ¿ C min{31;41; T1} h , and so the condition number of ATA is O( −2h−2).This completes the proof. In what follows, let (u; U;p) ∈ V × S × Q and (uh; Uh;ph) ∈ Vh × Sh × Qh be the solutions of problems (2.6) and (4.3), respectively. Lemma 4.3. We have the following error equation: for all (vh; Vh;qh) ∈ Vh × Sh × Qh; Bh((u; U;p) − (uh; Uh;ph); (vh; Vh;qh))=0: (4.20) Proof. It can be easily seen that the exact solution (u; U;p) of problem (2.6) satisÿes Bh((u; U;p); (vh; Vh;qh)) = Lh((vh; Vh;qh)) (4.21) for all (vh; Vh;qh) ∈ Vh × Sh × Qh which; combining with (4.3); implies (4.20). Lemma 4.4. The bilinear form Bh(·; ·) satisÿes the following inequalities 1=2 1=2 |Bh((v; V;q); (w; W;r))| 6 K ((v; V;q))K ((w; W;r)) (4.22) for all (v; V;q); (w; W;r) ∈ V × S × Q. Proof. Using the deÿnition of the bilinear form Bh(·; ·); the HUolder inequality; and m m 1=2 m 1=2 1=2 (LiMi) 6 Li Mi i=1 i=1 i=1 + for all L1;:::;Lm;M1;:::;Mm ∈ R ; we have Bh((v; V;q); (w; W;r)) T(h) 1 = (− (∇·V)T + ∇q)d · (− (∇·W)T + ∇r)d h h h k=1 k k k 2 + (V −∇vT)d · (W −∇wT)d h h h k k k 2 + (∇·v + q)d × (∇·w + r)d h h h k k k S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 69 2 + (∇trV + ∇q)d · (∇trW + ∇r)d h h h k k k 2 + (∇×V)d · (∇×W)d ; h h h k k k T(h) 1 d 6 |(− (∇·V)T + ∇q) |×1d h i h k=1 k i=1 k T × |(− (∇·W) + ∇r)i|×1d h k 2 2 d + |(V −∇vT) |×1d |(W −∇wT) |×1d h i i h h k i=1 k k 2 + |(∇·v + q)|×1d |(∇·w + r)|×1d h h h k k k 2 d + |(∇trV + ∇q) |×1d |(∇trW + ∇r) |×1d h i i h h k i=1 k k 2 2 2d −3d + |(∇×V) |×1d |(∇×W) |×1d h i i h h k i=1 k k T(h) d T T 6 (− (∇·V) + ∇q)i h (− (∇·W) + ∇r)i h 0;k 0;k k=1 i=1 d2 2 T T + (V −∇v )i h (W −∇w )i h 0;k 0;k i=1 2 + ∇ · v + q h ∇ · w + r h 0;k 0;k d 2 + (∇trV + ∇q)i h (∇trW + ∇r)i h 0;k 0;k i=1 2d 2−3d 2 + (∇×V)i h (∇×W)i h 0;k 0;k i=1 T(h) T T 6 − (∇·V) + ∇q h − (∇·W) + ∇r h 0;k 0;k k=1 70 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 T T + (V −∇v ) h (W −∇w ) h 0;k 0;k + (∇·v + q) h (∇·w + r) h 0;k 0;k + (∇trV + ∇q) h (∇trW + ∇r) h 0;k 0;k + (∇×V) h (∇×W) h 0;k 0;k 6 K1=2((v; V;q))K1=2((w; W;r)): This completes the proof. We are now in the position to prove our main result. Using Lemmas 4.2, 4.3, and 4.4 with the standard procedure similar to that in the usual ÿnite element analysis, we have the following H 1-optimal error estimate 2 d2+d+1 Theorem 4.2. Let (u; U;p) ∈ [V×S×Q]∩[H ()] and (uh; Uh;ph) ∈ Vh ×Sh ×Qh; h¡h0; be the solutions of problems (2.6) and (4.3); respectively. Then we have u − uh1; + U − Uh1; + p − ph1; 6 Ch( u2; + U2; + p2;): (4.23) Proof. From the orthogonality property (4.20); we can assert that Bh((u; U;p) − (vh; Vh;qh)+(vh; Vh;qh) − (uh; Uh;ph); (wh; Wh;rh))=0 for all (vh; Vh;qh); (wh; Wh;rh) ∈ Vh × Sh × Qh.Now choose (wh; Wh;rh)=(uh; Uh;ph) − (vh; Vh;qh) ∈ Vh × Sh × Qh so that Bh((uh; Uh;ph) − (vh; Vh;qh); (uh; Uh;ph) − (vh; Vh;qh)) =Bh((u; U;p) − (vh; Vh;qh); (uh; Uh;ph) − (vh; Vh;qh)) for all (vh; Vh;qh) ∈ Vh × Sh × Qh.Then ; for h¡h0; it follows from (4.12); (4.22); and (2.11) that 2 2 2 2 2 uh − vh1; + Uh − Vh1; + ph − qh1; 2 2 2 2 2 1=2 6 C( u − vh1; + U − Vh1; + p − qh1;) 2 2 2 2 2 1=2 ×( uh − vh1; + Uh − Vh1; + ph − qh1;) for all (vh; Vh;qh) ∈ Vh × Sh × Qh; and hence uh − vh1; + Uh − Vh1; + ph − qh1; 6 C( u − vh1; + U − Vh1; + p − qh1;) S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 71 for all (vh; Vh;qh) ∈ Vh × Sh × Qh.Therefore ; we get u − uh1; + U − Uh1; + p − ph1; 6 u − vh1; + U − Vh1; + p − qh1; + uh − vh1; + Uh − Vh1; + ph − qh1; 6 C( u − vh1; + U − Vh1; + p − qh1;) for all (vh; Vh;qh) ∈ Vh × Sh × Qh.Now ; choosing vh ∈ Vh; Vh ∈ Sh; and qh ∈ Qh such that (3.4); (3.5); and (3.6) hold when v; V; and q are replaced by u; U; and p; respectively; we obtain estimate (4.23). This completes the proof. 5. An H 1-equivalent error estimator In this section we brieIy introduce an [H 1()]d2+d+1-equivalent error estimator.Let ( u; U;p) ∈ V× S × Q and (uh; Uh;ph) ∈ Vh × Sh × Qh be the solutions of problems (2.6) and (4.3), respectively. For each h¡h0, deÿne 2 T 2 2 T 2 Eh = f − (− (∇·Uh) + ∇ph)0; + 0 − (Uh −∇uh )0; 2 2 2 2 + g − (∇·uh + ph)0; + ∇g − (∇trUh + ∇ph)0; 2 2 + 0 − (∇×Uh)0;: Then Eh is a computable quantity, and 2 T 2 2 T 2 Eh = − (∇·(U − Uh) )+∇(p − ph)0; + (U − Uh) −∇(u − uh) 0; 2 2 2 2 + ∇ · (u − uh)+(p − ph)0; + ∇tr(U − Uh)+∇(p − ph)0; 2 2 + ∇ × (U − Uh)0;: Thus, from (2.11), we have 2 2 2 2 1=2 C1( u − uh1; + U − Uh1; + p − ph1;) 2 2 2 2 1=2 6 Eh 6 C2( u − uh1; + U − Uh1; + p − ph1;) : 1 d2+d+1 Hence Eh is an [H ()] -equivalent a posteriori error estimator. 6. Concluding remarks In this investigation, we study a ÿnite volume scheme for approximating the solution of the generalized Stokes problem recast in the velocity–stress–pressure ÿrst-order system formulation by using continuous piecewise linear polynomials.This method is composed of a direct cell vertex ÿnite 72 S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 volume discretization step and an algebraic least-squares step.An optimal rate of convergence in the H 1 norm for all the unknowns, velocity, stress, and pressure, is achieved. The solution of the Navier–Stokes problem is sometimes approximated by a sequence of linear Stokes problems.Another often more adequate linearization of the Navier–Stokes equations is the Oseen problem [14] which o Similar to system (2.6), formulating this nonlinear second-order problem as a ÿrst-order system in terms of u, U, and p, we have − (∇·U)T +(u ·∇)u + ∇p = f in ; U −∇uT = 0 in ; ∇·u =0 in ; ∇trU = 0 in ; (6.2) ∇×U = 0 in ; u = 0 on @; n × U = 0 on @: Next, let (u0; U0;p0) be the solution of the Stokes problem (2.6). For n =0; 1; 2;:::, consider the following sequence of linear problems − (∇·Un+1)T + Anun+1 + ∇pn+1 = f in ; Un+1 −∇(un+1)T = 0 in ; ∇·un+1 =0 in ; ∇trUn+1 = 0 in ; (6.3) ∇×Un+1 = 0 in ; un+1 = 0 on @; n × Un+1 = 0 on @; where the coeHcient matrix An is given by @un @un 1 ··· 1 @x1 @xd . . An = . . : . . @un @un d ··· d @x1 @xd S.-Y. Yang / Journal of Computational and Applied Mathematics 147 (2002) 53–73 73 Now, for each stage n, n ¿ 0, we solve the corresponding linear problem by using the ÿnite volume scheme developed in Section 3.Since the velocity u is approximated by using piecewise linear n polynomials with respect to the triangulation Th, the coeHcient matrix A , n ¿ 0, is a piecewise n n n constant matrix at each stage.One might expect the sequence {(uh; Uh;ph)} generated by the above iterative process converging to a solution of the velocity–stress–pressure Navier–Stokes problem (6.2) as n →∞and h → 0.This issue has become the subject of current research of the author. 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