2 Jian Shen fundamental law in porous media ow. f is the source term. In general, the conductivity term is discontinuous for heterogeneous reservoirs

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1 MIXED FINITE ELEMENT METHODS ON DISTORTED RECTANGULAR GRIDS 1 Jian Shen Communicated by R. E. Ewing, R. D. Lazarov and J. M. Thomas. Institute for Scientic Computation Texas A & M University College Station, TX , U.S.A. July 1994 AMS(MOS) subject classications. 65N12, 65N3 Abstract A new mixed nite element method on totally distorted rectangular meshes is introduced with optimal error estimates for both pressure and velocity. This new mixed discretization ts the geometric shapes of the discontinuity of the rough coecients and domain boundaries well. This new mixed method also enables us to derive the optimal error estimates and existence and uniqueness of Thomas's mixed nite elements method on distorted rectangular grids [19]. The lowest order Raviart-Thomas mixed nite rectangular element method becomes a special case of both methods, when all the elements are degenerated to parallelograms. 1. Introduction Let be a bounded convex polygonal domain in R 2, with the = ; 1 [ ; 2, and ; 1 \ ; 2 =. We consider the homogeneous Dirichlet-Neumann boundary elliptic problem: ;r (rp) = f in p = on ; 1 rp n = on ; 2 (1:1) where is a 2 2 symmetric positive denite matrix, which is uniformly bounded below and above in n is the outward unit normal Extension to inhomogeneous boundary condition problems is straight forward. If we introduce a dependent vector valued variable u = ;rp, then, (1.1) is equivalent to the following rst order partial dierential equation system ;1 u = ;rp in (a) ru = f in (b) p = on ; 1 (c) u n = on ; 2 : (d) (1:2) In the simulation of uid ow in porous media, such as groundwater contamination and petroleum reservoir simulation, (1.2) arises frequently in a system of partial dierential equations. Here, p stands for the hydraulic pressure or pressure and u stands for the uid velocity or Darcy velocity. (1.2a) represents Darcy's law and (1.2b) is the mass conservation law, which is one of the 1 This research was supported in part by the Department of Energy under Contract No. DE{FG5{92ER25143, through the Institute for Scientic Computation, Texas A & M University. 1

2 2 Jian Shen fundamental law in porous media ow. f is the source term. In general, the conductivity term is discontinuous for heterogeneous reservoirs and the shapes of the discontinuous lines can be arbitrary. For example, in the simulation of incompressible miscible displacement, the conductivity can be written by = k, where k is a tensor representing the permeability of the medium which is discontinuous due to the the heterogeneity of the media and represents the viscosity of the uid. is a continuous function of both time and space variables, but may have a very sharp frontal change of values. According to the mass conservation law (1.2b), the velocity u must be continuous along the normal direction of a subdomain boundary, no matter whether is continuous or not. Roughly speaking, the discontinuity of the permeability cancels the discontinuity of the normal directional derivatives of the pressure along the subdomain boundaries. The mixed nite element discretizations provide reasonable approximations for such kind of problems but not the standard Galerkin type nite element ones, see Falk and Osborn [1]. In [7][8][17][18], on orthogonal non-uniform rectangular grids, superconvergence error estimates for both pressure and velocity are derived under certain local smoothness assumptions. Mixed - nite element methods on orthogonal grids and triangular grids have been studied intensively by many colleagues. Surveys on the development can be found, e.g., [3][13][15]. According to our computational experience on orthogonal grids, no approximation property can be achieved if the discontinuous lines fall inside any elements. Therefore, simulation of heterogeneous reservoirs require distorted rectangular grids to give an accurate model of the reservoir geology with a moderate number of grid cells. The mixed triangular nite element discretizations can approximate these discontinuous coef- cient problems well [1]. But computational implementation on unstructured triangular grids is not ease, especially in three-dimensional case, not mention that so many unknowns associated with triangular cells. So even though mixed triangular nite element methods was introduced at the same time as orthogonal rectangular nite element methods [14], the former is not as popular as the latter. There have been some considerations for mixed nite element methods on distorted rectangular grids. Thomas [19] dened non-conrming velocity spaces without showing stability and approximation properties. Then, either element-wise mass balance may not be guaranteed on any distorted rectangular elements or the divergence of the discrete velocity spaces are not in the corresponding discrete pressure spaces. Farmer et. al. [11] and Russell [16] using Thomas's denition have obtained very encouraging computational results. In [16], using a nite volume approach by means of Thomas's nite element velocity space denition on distorted grids, Russell obtained superconvergence for both pressure and velocity in computational experiments. In [11][16], the velocity space is continuous in the normal direction on the element interfaces, thus, mass balance property is retained element-wise, but the divergence of the velocity space is no longer in the pressure space. Then, Russell [16] conjectured that the necessary and sucient condition for mixed methods B-B inf-sup condition to be satised only in the limit sense: when h tends to zero, all the irregular elements are almost parallelograms. In this paper, we develop a mixed nite element method which is a natural and compact generalization of the lowest order Raviart-Thomas mixed nite orthogonal rectangular element method with stability and optimal error estimates. This method can be extended to three-dimensional case. Due to the complexity of the shapes of distorted cubes, we shall discuss three-dimensional case in another paper [5]. By our new mixed nite element discretization, we then enable to prove the Thomas's method on distorted grids [19] and Russell's conjecture about the B-B inf-sup condition

3 Mixed Methods on Distorted Grids 3 [16], if h is suciently small. Comparing with the lowest order mixed triangular discretization, we reduce the number of unknowns signicantly on relatively well structured grids with the some order of accuracy. We feel that these two are practical methods for both numerical mathematicians and engineers from either theoretical or application point of views. To the author's knowledge, this is the rst time to introduce such a method with solid analysis. According to [2], Professor Thomas has a way of proof for this numerical method in a totally dierent approach but he has never published his result before. The remainder of this paper is organized as follows: We dene the weak form of (1.2) in x2 with some notations for Sobolev spaces, the partition of the domain and some assumptions. In x3, we list some of the properties of bilinear and the contravariant Piola transformations. The mixed nite element spaces are dened in x4. In x5, projections and approximation properties of projections are discussed. The error estimates are presented in x6. The theoretical support for Thomas's denition [19] and Russell's conjecture about the B-B inf-sup condition [16] is x7. Finally, in x8, we give some conclusion remarks for implementations. 2. The Weak Formulations and Assumptions Denote by ( ) the usual L 2 () or (L 2 ()) 2 inner product. The spaces H k (), for k a positive integer, will be the usual Hilbert spaces X equipped with X the norms kk 2 k = k@ k 2 = ): (2:1) jjk jjk The H(div ) is dened by n o H(div ) v 2 (L2 ()) 2 : rv 2 L 2 () equipped with the graph norm kvk 2 H(div) = kvk2 + kr vk 2 : (2:2) The Sobolev spaces W k 1 (), for k a positive integer, are equipped with the norms Denote kk k 1 = max k@ k 1 jjk kk 1 = ees sup jj: V = fv 2 H(div ) : v n = on ; 2 g ( L W = 2 () if ; 1 6= L 2 ()nr if ; 1 = : For solution of (1.2) and v 2 V, applying the Green's formula and the homogeneous boundary condition, we obtain that ;(rp v) =(p rv) pv nds =(p rv): The mixed weak form of (1.2) is to seek a pair (u p) 2 V W such that (2:3) ( ;1 u v) = (p rv) v 2 V (ru w) = (f w) w 2 W: (2:4)

4 4 Jian Shen To dene a nite element method, we need a partition T h of into element. For 2T h, is distorted rectangular but all its four edges are straight lines (see Figure 1.). If any edges of, say, e, is on the either Dirichlet or Neumann condition is imposed but not the both. Denote by jj the area of. We require the quasi-regularity of the mesh, i.e., there are 1 >h> and C C > independent ofh such that C h 2 jj C h 2 8 2T h : (2:5) Note that we require that 2 T h be convex to insure that the Jacobians (dened in x3) are no-singular. Through out the rest of this paper, C will be used to stand for a positive constant independent ofh. We assume that the boundaries of dierent rocktypes or regions of dierent permeability, i.e., the discontinuous lines are combination of nite many straight lines so that we can make grid lines coincide with the discontinuous lines. Denote by ; the set of points where the coecient is discontinuous, adopted from [7][8][17][18] we make the following assumptions for solutions of (2.4). Assumption ; consists of only line segments such that a distorted meshcan be formed on to locate ; on grid lines. 2. All the entries of are smooth in n;, say,in W 1 1 (n;). 3. the pressure solution p 2 H 1 () T H 2 ((n;). The one side normal and tangential derivatives of p on ; are well dened and belong to L 2 (;), such that the velocity, u is continuous in the normal direction of ;. In addition, we require that u 2 (H 1 ()) 2 and ru 2 H 1 (). y 6 ^x 4 ^y 6 ^x 3 ^x 1 - ^x ^x 2 F - H H H H H Hx 3 C CC C CC x 1 x 4 Figure 1. The bilinear mapping. x 2 - x 3. Properties Related to a Change of Variables and the Bilinear Transformation Denote x =(x y) t ^x =(^x ^y) t :

5 Mixed Methods on Distorted Grids 5 Dene We may also use f x x )t )t : to stand for the partial derivative off to the variable x. Let x i =(x i y i ) t, i = be the four vertices of element counted anti-clockwise, and ^x i, i = be the four vertices of the unit square (see Figure 1.), ^x 1 =( ) t ^x 2 =(1 ) t ^x 3 =(1 1) t ^x 4 =( 1) t : (3:1) Let x ij = x i ; x j, the bilinear transformation, F :, is satisfying x = F(^x) =x 1 + x 21^x + x 41^y +(x 32 ; x 41 )^x^y (3:2) x i = F(^x i ) i = : (3:3) Denote by J the correspondent Jacobi matrix, and J = j det Jj, the absolute value of the Jacobian, J = x^x y^x x^y y^y J = j det Jj= x^x y^y ; x^y y^x > (3:4) and J ;1 = ^x x ^x y ^y x ^y y = 1 J y^y ;x^y ;y^x x^x : (3:5) Note that, under the bilinear transformation (3.2), J is a linear function of ^x and ^y, J = (x 21 y 41 ; x 41 y 21 )+(x 21 y 32 ; x 32 y 21 + x 41 y 21 ; x 21 y 41 )^x +(x 32 y 41 ; x 41 y 32 )^y = 2[ 124 +( 123 ; 124 )^x +( 234 ; 123 )^y] = + ^x + ^y: (3:6) In (3.6), ijk is the area of the triangle of x i x j x k. The area of element is 1 1 Jd = Jd^xd^y = : (3:7) jj = d = by The Piola transformation P, [3] [19], associated with the change of the variables (3.2) is dened u = P ^u = 1 J ^u: (3:8) J By the chain rule, Since rp = J ;t ^r^p ^p = p(f(^x)): (3:9) fd = ^fjd ^f = f(f(^x)) (3:1)

6 6 Jian Shen letting v = 1 J ^v, wehave J rp vd = ^r^p ^vd : (3:11) The following useful properties about Piola transformation can be found in [3] [15] rv = 1 J rvd = v nds = Let the 2 2 positive denite matrices and satisfy that ^r^v v = 1 J ^v: (3:12) 1 ^r^vd v = J ^v: (3:13) J ^v ^nd^s v = 1 J ^v: (3:14) J = JJ ;1 J ;t (3:15) then, = 1 J J J t ;1 = 1 J J t ;1 J and ;1 = JJ ;t ;1 J ;1 : If then, (3.8), (3.9) and (3.15) yield ^u = ; ^r^p (3:16) u = P ^u = ;rp: (3:17) Let us investigate the simple orthogonalities of Piola(under bilinear) transformation. The normal direction to the level curve, ^x =^x(x y) = constant, in the xy-plane is n^x = Jr^x = J(^x x ^x y ) t =(y^y ;x^y ) t (3:18) and the normal direction to the level curve, ^y =^y(x y) = constant, in the xy-plane is n^y = Jr^y = J(^y x ^y y ) t =(;y^x x^x ) t : (3:19) Similarly, the unit normals of the level curves, ^x = constant and ^y = constant in the ^x^y-plane are ^n^x = ^r^x = 1 respectively. According to (3.9), we have and ^n^y = ^r^y = 1 Let u = P ^n^x and v = P ^n^y, then, at any point (x y) 2, (3:2) n^x = JJ ;t^n^x and n^y = JJ ;t^n^y : (3:21) u n^y = ^n^x ^n^y = and v n^x = ^n^y ^n^x =: (3:22)

7 Mixed Methods on Distorted Grids 7 Let e i+1 be the line segment, x i x i+1, and je i+1 j = jx i i+1 j, be the length of x i x i+1, i = Let e 1 be the line segment, x 4 x 1, and je 1 j = jx 41 j. Then, the corresponding edges to e i i = of are ^e 1 = f(^x ^y) j ^x = ^y 1g ^e 2 = f(^x ^y) j ^y = ^x 1g (3:23) ^e 3 = f(^x ^y) j ^x =1 ^y 1g ^e 4 = f(^x ^y) j ^y =1 ^x 1g: Let ^n i be the unit normal of ^e i, i.e., ^n i = (1 ) t i = 1 3, and ^n i = ( 1) t i = 2 4. Direct culculations of (2.18) and (2.19) lead to jn^x j^x= = je 1 j jn^y j^y= = je 2 j jn^x j^x=1 = je 3 j jn^y j^y=1 = je 4 j (3:24) then, the unit normal, n i,ofe i, i = , are expressed by n 1 = n^xj^x= je 1 j n 2 = n^yj^y= je 2 j n 3 = n^xj^x=1 je 3 j n 4 = n^yj^y=1 : (3:25) je 4 j Actually, ;n 1 ;n 2 n 3 n 4, are the unit outward normals of. Denition (3.25) is convenient to dene the unique nodal velocity values. Later, with an obuse of notation, we shall also use (3.25) to represent the unit noutward ormals of and keep the sign dierences in mind. Lemma 3.1. then, For the bilinear transformation (3.2) and the Piola transformation (3.8), if v = 1 J J ^v, v n i ds = ^v ^n i d^s i = : (3:26) e i ^e i Proof Note that by a change of variable on e i, ds = je i jd^s i = According to (3.18), (3.19), (3.21), (3.24) and (3.25), je i jv n i = 1 J J ^v (JJ ;t^n i )=^v ^n i i = : 4. The Mixed Finite Element Spaces In [19], page IX-21, if the elements are distorted rectangles, Thomas denes the discrete velocity space by imposing continuity in the normal direction at the mid-point of each element boundary edge with the same degree of polynomials as in the case of orthogonal rectangular elements. No stability or error estimates are discussed in [19] for this case. Actually, extra care has to be taken, otherwise, either the discrete velocity space may no longer be guaranteed a subspace of V or the divergence of the discrete velocity space is no longer a subspace of the discrete pressure space. In this section we generate a pair of discrete velocity space, V h and pressure space, W h, such that V h V H(div ) (4:1) divv h W h W L 2 () are satised. We shall keep the same number of unknowns as in [19], therefore, our mixed nite element method is in consistence with the lowest-order Raviart-Thomas orthogonal rectangular mixed nite element method [14][19]. That is, when the elements degenerate back to parallelograms, our mixed nite element method is exactly the lowest-order Raviart-Thomas orthogonal rectangular mixed nite element method. Most importantly, because of (4.1) our new mixed nite element 2

8 8 Jian Shen method satises the B-B inf-sup condition which is necessary and sucient condition for existence and uniqueness (e.g., see [3] [13]). In x7, by means of our new mixed nite element method, we shall show the convergence, existence and uniqueness of Thomas's method. The pressure space W h : Denote () the characteristic function of domain, then, W h = spanf()g 2Th. W h is the piecewise constant space, then, W h L 2 (). The velocity space V h : In order to insure the normal direction continuity on the element edges, we require that the restriction of the nodal bases b i 2 V h, i = on element satisfy rb i = constant b i n j j ej = ij : (4:2) By (3.12) and the orthogonality property of the Piola-bilinear transformation (3.22), (4.2) is equivalent to ^r^b i = J constant (4:3) ^b i ^n j j^ej = je i j ij : Note that J is a linear function of ^x and ^y by (3.6). Lemma 4.1. There exists a ^b 2 R 2 dened on such that ^r^b = J jj ; 1 ^b ^n j j^ej = j = (4:4) and then, ^r^b d =: (4:5) Proof Indeed, let ^b = = ^x(^x ; 1) = 2jj ^y(^y ; 1) (4:6) 2jj then, the ^b given by (4.6) satises (4.4) and (4.5). In addition, maxfjj jjg < 1 4 : (4:7) 2 >From (4.6), we see that ^b is solely determined by J. Now, let ^b 1 = je 1 j ^b 2 = je 2 j 1 ; ^x 1 ; ^y ;je 1 j^b ;je 2 j^b ^b 3 = je 3 j ^b 4 = je 4 j ^x ^y + je 3 j^b + je 4 j^b : (4:8) Then, let b i = 1 J J ^b i i = : (4:9)

9 Mixed Methods on Distorted Grids 9 It is straight forward to check that those nodal bases dened by (4.4) and (4.6) satisfy (4.3). Hence, the nodal bases given by (4.8) satisfy (4.2). Therefore, any u 2 V h, its restriction on is uniquely determined by 4X uj = i=1 u i b i u i = u n i j ei i = : (4:1) Clearly, divv h W h, and from the dimension of V h and W h, the operator div : V h W h is onto, therefore W h divv h. Finally, thenumerical space V h W h satises (4.1) as we desired. Remark 4.1. From (4.6) and (4.7) we can see that the ^x-components of ^b i, i = are quadratic functions of ^x and the ^y-components of ^b i, i = are quadratic functions of ^y. If the we drop the quadratic terms, the ux at all the boundary edges of are the same, but unfortunately the divergence of the discrete velocity space will not be the piecewise constant pressure space any more.we shall overcome this problem in x7. Now, we are ready to dene the approximation problem for (2.4). The numerical scheme to approximate (2.4) is to seek a pair (u h p h ) 2 V h W h such that ( ;1 u h v) = (p h rv) v 2 V h (ru h w) = (f w) w 2 W h : (4:11) 5. The Projections For mixed nite element methods, projections play dominant roles in stability and error estimates, according to Fortin's Lemma [12]. First, the L 2 -projection is dened in the usual sense, Q h : W W h,by (f ; Q h f w)= (f ; Q h f)wd = 8 f 2 W w2 W h : (5:1) By denition (5.1), we can see that, if f is a piecewise constant function, f Q h f, hence, applying Bramble-Hilbert Lemma [2] and the quasi-regularity (2.5), we have that kf ; Q h fk Ckfk 1 h: (5:2) The H(div)-projection is an extension [18] of the Raviart-Thomas projection [14][19], h : V V h, given by r(u ; h u)rvd = 8 u 2 V v 2 V h : (5:3) (r(u ; h u) rv) = If we take rv = w 2 W h in (5.3) and combining (5.1), we derive that that is, div h = Q h div. r h u wd= ru wd= Q h ru wd (5:4)

10 1 Jian Shen Now, we give the local expression for h u explicitly for u 2 (H 1 ()) 2. Since h u 2 V h, the restriction of h u on is given by h uj = 4X i=1 u i b i = P In (5.3), taking v 2 V h such thatrv = () 2 W h, = r(u ; h u) ()d = r(u ; h u)d = = r(^u ; ^ h^u)d = (^u ; ^ h^u) ^nd^s: By Lemma 3.1., if we dene ^u i = ^e 4X i=1 ^u ^n i d^s, i = , then, u i = 1 u n i ds = ^u i je i j e i je i j = 1 je i j ^u i je i j ^b i Def = P ^ h^uj : (u ; h u) nds (5:6) ^e i ^u ^n i d^s i = (5:7) therefore, h uj = P ^ h^uj satisfy (5.6). Then, (5.5) can be rewritten as ^ h^uj = ^u 1 + g 1^x + g ^u 2 + g 2^y + g (5:8) where g 1 =^u 3 ; ^u 1 = g 2 =^u 4 ; ^u 2 ^u^y d g =^u 3 ; ^u 1 +^u 4 ; ^u 2 = ^r^ud = rud: >From Cauchy-Schwarz inequality and (2.5), q jg j = j rudj jjkr uk Chkr uk : (5:1) We are ready to prove (5:9) Lemma 5.1. Assume that u 2 (H 1 ()) 2,andru 2 H 1 (), then, ku ; h uk Ckuk 1 h kr (u ; h u)k Ckr uk 1 h: (5:11) Proof The second estimate in (5.11) is actually (5.2). For the rst one, we get ku X ; h uk 2 = ku X ; h uk 2 = 2T h 2T h (u ; h u) (u ; h u)d

11 Mixed Methods on Distorted Grids 11 and (u ; h u) (u ; h u)d = 1 J (^u ; ^ h^u) t J t J (^u ; ^ h^u)d : By the Bramble-Hilbert Lemma [2], it suces to assume that each component of^u is at most linear, i.e, ^u = B ^u^x +(^x ; 1 )@^x^u^x +(^y ; 1 )@^y 2 2 ^u^y +(^x ; 1 )@^x^u^y +(^y ; 1 )@^y ^u^y 2 2 where ^u^x =^u^x ( )and^u^y =^u^y ( ). Then by (5.7), 2 1 C A ^u 1 =^u^x ; 1 ^u 2 =^u^y ; 1 ^u^y : Denote ^v = ^u ; ^ h^u = (^y ; 1 2 )@^y ^u^x ; g (^x ; 1 2 )@^x^u^y ; g 1 C A v = P ^v: According to [4] the local estimates for the bilinear map, the quasi-regularity assumption (2.5), (4.7) and (5.1), we have kvk Chkuk 1 : Summing over all the element 2T h,we nish the proof of Lemma Thus, if the domain is a convex polygon, so that H 2 ()-regularity iswell dened, by Fortin's Lemma [12], the B-B inf-sup condition (w rv) kwk C sup 8 w 2 W h (5:12) v2v h kvk H(div) holds for V h W h dened in x4. Further, since the mass balance property is the property of H(div ), and, V h V H(div ), the mass balance property is retained for V h element-wise. Remark 5.1. According to [12][2], divv h W h is not a necessary condition for (5.12) being held, which isalsoveried by the result of x7 in this paper. 6. Approximation Properties Taking advantage of the well settled machinery for analyzing mixed schemes, by inequality (5.2) and Lemma 5.1., we have that Theorem 6.1. Under the assumption 2.1., let the pair (u p) be the solution of (2.4) and (u h p h ) be the solution of (4.6). Then, ku ; u h k Ckuk 1 h kr (u ; u h )k Ckr uk 1 h kp ; p h k C(kpk 1 + kuk 1 ) h: (6:1)

12 12 Jian Shen Proof First, we showthat The error equations of (2.4) and (4.11) take the form of ku ; u h k Cku ; h uk : (6:2) ( ;1 (u ; u h v) = (p ; p h rv) v 2 V h (r(u ; u h ) w) = w 2 W h : (6:3) Since is bounded, (6.3) yields 1 C ; u hk 2 ( ;1 (u ; u h ) u ; u h ) = ( ;1 (u ; u h ) u ; h u)+( ;1 (u ; u h ) h u ; u h ) = ( ;1 (u ; u h ) u ; h u): For = r( h u ; u h ) 2 W h, and the L 2 -projection (5.1), it is straightforward to check that ( ;1 (u ; u h ) h u ; u h )=. Thus, (6.2) follows by applying Cauchy-Schwarz inequality, hence, Lemma 5.1 gives rise to the rst inequality in (6.1). Again, because = r( h u ;u h ) 2 W h,(5.3) and (6.3) imply that kr (u ; u h )k kr(u ; h u)k : (6:4) Then, inequality (5.11) yields the second one of (6.1). Finally, applying the discrete B-B inf-sup condition (5.12), (Q kq h p ; p h k C sup h p ; p h rv) v2v h kvk H(div) = C sup v2v h ( ;1 (u ; u h ) v) kvk H(div) Cku ; u h k : Combining above results with (6.2), the proof is complete. 2 By the duality argument of the standard mixed nite element methods(e.g., see [3]), we can derive the following superconvergence result for pressure alone Corollary 6.2. Under the assumption of Theorem 6.1., then, kq h p ; p h k C(kuk 1 + kr uk 1 )h 2 : (6:5) 7. On Thomas's Velocity Space Denote by ~V h the discrete velocity space on the same partition T h dened by Thomas [19]. In this section, we overcome the diculty of Thomas's velocity space: div ~V h is not a subspace of W h and then, give theoretical support for the usage of Thomas's method. First, we set up some notations. From (4.8), if we dene then, for ^b given by (4.6),let ^t i = ^b i ;je i j^b i =1 2 ^t i = ^b i + je i j^b i =3 4 (7:1) b = P ^b t i = P ^t i i = : (7:2)

13 Mixed Methods on Distorted Grids 13 For any ~u 2 ~V h, its restriction on is uniquely determined by ~uj = 4X i=1 ~u i t i ~u i = ~u n i j ei i = : (7:3) Since is a unit square, the linear combination of ^t i, i = are tensor product of onedimensional linear polynomials of ^x and ^y respectively. By denition [14][19], ^t i, i = are the bases for lowest-order Raviart-Thomas mixed nite orthogonal rectangular element bases on the reference element. But as long as P is not a constant matrix, t i, i = do not satisfy (4.2). Therefore, div ~V h 6= W h,even though ~V h V. The following lemma tells us how close the two spaces V h and ~V h are related. Lemma 7.1. nodal values, i.e., on the boundaries of each element, Further, if we let then, for u 2 V h, For any u 2 V h, there is unique ~u 2 ~V h such that u and ~u have exactly the same (u ; ~u) n j = i = T h : (7:4) u = u ; ~u (7:5) ku ; ~uk = ku k Ckr uk h kr (u ; ~u)k = kr u k Ckr uk h: Proof For u 2 V h, u i = u n i j ei is a constant, then, combining with Lemma 3.1., u i = 1 u n i ds = 1 ^u ^n i d^s: je i j e i je i j ^e i (7:6) Let ^u i = je i ju i then, by (5.8) and (5.9), u = X 2T h = X 2T h P Def = ~u + u : 4X u i b i i=1 4X i=1 ^t i ^u i je i j = X 2T h P + X 4X i=1 ^b i ^u i je i j b rud 2T h (7:7) By denition (4.1), (7.3) and the nodal bases relation (7.1), the rst assertion (7.4) is proved. Now, from (2.5), (3.2), (3.4) and (4.7), we have jjjb j = p b b jj 4J q (x^x + x^y ) 2 +(y^x + y^y ) 2 Ch:

14 14 Jian Shen Then, For (7.6), from (7.7), and from (3.12), Thus, for jj 2 ku k 2 = X (J ;jj) 2 d = 1 2T h Ch 2 X ru j = rb = 1 J (rb ) 2 d = 1 = (2 + 2 ) 12 2 rud b b d 2T h kr uk 2 = Ckr uk 2 h2 : rud rb ^r^b = 1 jj (J ;jj) 2 <J =minfjg Ch 2 and jj jj Ch 2. Finally, J 1 ; jj J : d ( ) 12J Ch 2 2 (^x ; 1 2 )2 + 2 (^y ; 1 2 )2 ; 2(^x ; 1 2 )(^y ; 1 2 ) d^xd^y kr u X k 2 = ( rud) 2 (rb ) 2 d 2T h X Ch 2 kr uk 2 = Ckr uk 2 h2 : 2T h 2 Apparently, Lemma 7.1. also implies that ~V h V. Now, we dene another approximation problem for (2.4). The numerical scheme to approximate (2.4) is to seek a pair (~u h p h ) 2 ~V h W h such that ( ;1 ~u h ~v) = (~p h rv) ~v 2 ~V h (7:8) (r~u h w) = (f w) w 2 W h : Combined with inequality (5.2) and Lemma 5.1., Lemma 7.1 enables us to show that Theorem 7.2. Under the assumption 2.1., let the pair (u p) be the solution of (2.4) and (~u h ~p h ) be the solution of (7.8). In addition, if h is suciently small, then, kp ; ~p h k C(kpk 1 + kuk 1 ) h ku ; ~u h k Ckuk 1 h kr (u ; ~u h )k Ckr uk 1 h: (7:9)

15 Mixed Methods on Distorted Grids 15 Proof The error equations of (2.4) and (7.8) take the form of ( ;1 (u ; ~u h ~v) = (p ; ~p h r~v) ~v 2 ~V h (r(u ; ~u h ) w) = w 2 W h : (7:1) Since is bounded, (7.1) yields 1 C ; ~u hk 2 ( ;1 (u ; ~u h ) u ; ~u h ) = ( ;1 (u ; ~u h ) u ; h u)+( ;1 (u ; ~u h ) h u ; ~u h ): (7:11) Let u be the O(h)-perturbation part of h u such that h u ; u 2 ~V h. Similarly, let u h be the O(h)-perturbation part of ~u h such that ~u h + u h 2 V h. Thus, Lemma 7.1. gives rise to ku k Chkr uk and ku h k Chkr uk. Rewrite the second term in (7.11) as Since ( ;1 (u ; ~u h ) h u ; ~u h )=( ;1 (u ; ~u h ) h u ; u ; ~u h )+( ;1 (u ; ~u h ) u ): j( ;1 (u ; ~u h ) u )jcku ; ~u h k ku k Cku ; ~u h k kr uk h and h u ; u ; ~u h 2 ~V h, the rst equation of (7.1) gives rise to ( ;1 (u ; ~u h ) h u ; u ; ~u h )=(p ; ~p h r( h u ; u ; ~u h )): Then, (5.4) and the second equation of (7.1) lead to j(p ; ~p h r( h u ; u ; ~u h ))j = j(p ; ~p h r(u h ; u )) + (Q h p ; ~p h ru )j Put the above results in (7.11), we derive that C (hkpk 1 + kq h p ; ~p h k ) kr uk h: ku ; ~u h k 2 C (ku ; ~u h k kuk 1 + kq h p ; ~p h k kr uk + hkpk 1 kr uk ) h: (7:12) Next, we show that, if h is suciently small, kq h p ; ~p h k C ku ; ~u h k + h 2 kpk 1 : (7:13) If v 2 V h is arbitrary, then, Lemma 7.1. implies that v = ~v + v, ~v 2 V ~ h such that kv k Chkr vk and kr v k Chkr vk. Since v 2 V h, (5.1) and the rst equation of (7.1) yield (Q h p ; ~p h rv) = (p ; ~p h rv) = (p ; ~p h r~v)+(p ; ~p h rv ) = ( ;1 (u ; ~u h ) ~v)+(p ; ~p h rv ): (7:14) Triangal inequality and Lemma 7.1. yield k~vk kvk + Chkr vk Ckvk H(div) and kr v k Ckr vk h Ckvk H(div) h:

16 16 Jian Shen Applying Cauchy-Schwarz inequality to (7.14), we have j(q h p ; ~p h rv)j C ku ; ~u h k + h 2 kpk 1 + hkq h p ; ~p h k kvk H(div) : (7:15) Then, (5.12) and (7.15) give rise to kq h p ; ~p h k C ku ; ~u h k + h 2 kpk 1 + hkq h p ; ~p h k and if h is suciently small, (7.13) holds. Again, if h is suciently small, (5.2), (5.11), (7.12) and (7.13) yield the rst two inequalities of (7.9). Finally, by (5.11) and the second equation in (7.1), kr u ; ~u h k 2 = (r(u ; ~u h ) r(u ; h u)) + (r(u ; ~u h ) r( h u ; ~u h )) = (r(u ; ~u h ) r(u ; h u)) + (r(u ; ~u h ) r~u h ) +(r(u ; ~u h ) r( h u ; ~u h ; ~u h )) = (r(u ; ~u h ) r(u ; h u)) + (r(u ; ~u h ) r~u h ) Ckr (u ; ~u h )k kr uk 1 h: Thus, Theorem 7.2. is proved. Remark 7.1. Throughout the argument of Theorem 7.1., we have tacitly assumed that (7.8) is solvable. The existence and uniqueness of (7.8) can be established from (7.12) and (7.13). Since for nite dimensional cases, uniqueness implies existence, we only need to demonstrate uniqueness. Momentarily, we interpret (~u h ~p h ) in (7.8) as a solution pair of the homogeneous problem. If h is suciently small, (7.13) yields and (7.12), with f =, shows that k~p h k Ck~u h k k~u h k =: Therefore, ~p h and ~u h vanish for small h. Finally, the necessity of the discrete B-B inf-sup condition for existence and uniqueness (e.g., see [13]) implies that the discrete B-B inf-sup condition holds for ~V h W h also, if h is suciently small. Similar to Corollary 6.2., we have the superconvergence result for pressure alone 2 Corollary 7.3. Under the assumption of Theorem 7.2., then, Proof Dene the adjoint problem of (1.2) kq h p ; ~p h k C(kuk 1 + kr uk 1 )h 2 : (7:16) ;1 q = ;r in rq = Q h p ; ~p h in = on ; 1 q n = on ; 2

17 Mixed Methods on Distorted Grids 17 and assume that kk 1 kqk 1 CkQ h p ; ~p h k. Let q be the O(h)-perturbation part of h q such that h q ; q 2 ~V h and kq k Chkr qk by Lemma 7.1. Then, kq h p ; ~p h k 2 = (Q h p ; ~p h rq) =(Q h p ; ~p h r h q)=(p ; ~p h r h q) = (p ; ~p h r( h q ; q )) + (p ; ~p h rq ) = ( ;1 (u ; ~u h ) h q ; q )+(p; ~p h rq ) = ( ;1 (u ; ~u h ) h q ; q)+( ;1 q u ; ~u h ) ; ( ;1 (u ; ~u h ) q )+(p; ~p h rq ) = ( ;1 (u ; ~u h ) h q ; q)+( r(u ; ~u h )) ; ( ;1 (u ; ~u h ) q )+(p; ~p h rq ) = ( ;1 (u ; ~u h ) h q ; q)+( ; Q h r(u ; ~u h )) ; ( ;1 (u ; ~u h ) q )+(p; ~p h rq ) C (kuk 1 kqk 1 + kr uk 1 kk 1 +(kuk 1 + kpk 1 kr qk ) h 2 + C 1 kq h p ; ~p h k kr qk h: If h is suciently small, then, (7.16) is valid Conclusion Remarks >From Theorem 6.1. and Theorem 7.2., we can see that these two mixed method discretizations (4.11) and (7.8) provide O(h)-order accurace for both pressure and velocity. The only shortcoming of (7.8) is requiring h be suciently small. This provides theoretical support for the computational results of Farmer et. al. [11] and partially for Russell's (not the superconvergence for velocity) [16]. The discrete space V h W h given in x4, not only serves as the theoretical foundation for both methods but also oers a reasonable solver to approximate (2.4). In computational implementations, compared with the space ~V h W h given in x7, the only draw back of method (4.11) is the small amount of work to calculate the integrals of the quadratic terms element-wise in the mass matrix formulation. The mass matrices of these two methods have exactly the same band structure. These two methods would be superior in accuracy, to the corresponding triangular method, if some superconvergence results for velocity can be derived, similar to [6][7][8][9] for orthogonal grid. But this is beyond the purpose of this paper. Finally, the piecewise constant Lagrange multipliers on the boundaries of elements are well dened for our mixed nite element space dened in x4 and the Thomas's [19] given in x7. Therefore, the superconvergence results of the Lagrange multipliers will follow without any diculties. The great advantage of introducing Lagrange multipliers is that we can convert the saddle-point problem (4.11) and (7.8) to positive denite algebraic systems. We refer to Arnold and Brezzi [1] for details on the orthogonal rectangular grids. References [1] D. N. Arnold and F. Brezzi, Mixed and nonconforming nite element methods: implementation, postprocessing and error estimates, R.A.I.R.O. Anal. Numer. 19(1985). [2] J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with application to Hermite interpolation, Numer. Math., 16 (1971), pp. 362{369.

18 18 Jian Shen [3] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, [4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, [5] R. E. Ewing, R. D. Lazarov, and J. Shen, Mixed nite element methods on logically cubical grids, (In preparation). [6] R. E. Ewing, R. D. Lazarov, and J. Wang, Superconvergence of the velocity along the Gauss lines in mixed nite element methods, SIAM J. Numer. Anal., 28 (1991), pp. 151{129. [7] R. E. Ewing, R. C. Sharpley and J. Shen, On accuracy of the lowest-order quadrilateral mixed nite element discretization in solving heterogeneous ow problems, (In preparation). [8] R. E. Ewing and J. Shen, A discretization scheme and error estimate for second-order elliptic problems with discontinuous coecients, preprint of ISC, TAMU [9] R. E. Ewing, J. Shen and J. Wang, Point distributed algorithms for second-order elliptic equations in mixed form, Proceedings of Fifth orean Advanced Institute for Science Technology International Workshop in Analysis, U. J. Choi, et. al. eds. Taejon, orea, 1993, pp. 53{76. [1] R. S. Falk and J. E. Osborn, Remarks on mixed nite element methods for problems with rough coecients, Math. Comp., vol. 62(25), 1994, pp. 1{19. [11] C. L. Farmer, D. E. Heath and D. O. Moody, A global optimization approach to grid generation, SPE 21236, Proceedings of 11th SPE Symposium on Reservoir Simulation, Anaheim, CA, Feb. 17-2, [12] M. Fortin, Analysis of the convergence of mixed nite element methods, R.A.I.R.O. Anal. Numer., 11 (1977), pp. 341{354. [13] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer- Verlag, [14] P. A. Raviart and J. M. Thomas, A mixed nite element method for 2nd order elliptic problems, Mathematics Aspects of the Finite Element Method, Lecture Notes in Mathematics 66, Springer-Verlag (1977) pp. 292{315. [15] J. E. Robert and J. M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, (P.G. Ciarlet and J.L. Lions, eds.), Vol. II, Finite Element Methods (Part 1), North- Holland, Amsterdam (1889). [16] T. F. Russell, Rigorous discretizations for irregular block-centered grids, Colloquium in the Institute for Scientic Computation, Texas A & M University, Texas, April [17] J. Shen, Mixed Finite Element Methods: Analysis and Computational Aspects, Ph.D. Thesis, University ofwyoming, [18] J. Shen, A block nite dierence scheme for second-order elliptic problems with discontinuous coecients, SIAM J. Numer. Anal (to appear).

19 Mixed Methods on Distorted Grids 19 [19] J. M. Thomas, Sur l'analyse numerique des methods d'elements nis hybrides et mixtes, These d'etat, Universite Pierre et Marie Curie, Paris (1977). [2] Personal discussion wiht Prof. J. M. Thomas. Department of Mathematics and Institute for Scientific Computation Texas A & M University College Station, TX shen@isc.tamu.edu