The Effect of Numerical Integration. in Galerkin Methods for Compressible. Miscible Displacement in Porous Media

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1 Nonlinear Analysis and Differential Equations, Vol. 5, 217, no. 1, HIKARI Ltd, The Effect of Numerical Integration in Galerkin Methods for Compressible Miscible Displacement in Porous Media Nguimbi Germain Ecole Nationale Supérieure Polytechnique Marien Ngouabi University, Brazzaville, Congo Pongui Ngoma Diogène Vianney and Likibi Pellat Rhoss Beauneur Department of Mathematics Marien Ngouabi University, Brazzaville, Congo Copyright c 216 Nguimbi Germain et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We consider the effect of numerical integration in finite element methods for nonlinear parabolic system describing a two component model for the single-phase, miscible displacement of one compressible fluid by another in a porous medium. We give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order L 2 -error estimates and almost optimal L -error estimates are derived when the imposed external flows are smoothly distributed.

2 18 Nguimbi Germain et al. Keywords: finite element, quadrature scheme, molecular diffusion and dispersion, error estimates 1 Introduction We analyze the effect of numerical integration for a problem arising in the single-phase, miscible displacement of one compressible fluid by another in a porous medium. The reservoir Ω is of unit thickness and is identified with a bounded domain in R 2. Let c i denote the volumetric concentration of the ith component of the fluid mixture i = 1,..., n. The state equation for the model is taken to be dρ i ρ i = z i dp, (1.1) where ρ i is the density of the ith component solely dependent on the pressure p and z i is the constant compressibility factor for the ith component. The Darcy velocity of the fluid mixture is given by u = k p, (1.2) µ where k = k(x) is the permeability and µ = µ(c 1,..., c n ) is the viscosity of the fluid. Assume that no volume change occurs in mixing the components and that a diffusion tensor D = D(x, u) representating molecular diffusion and dispersion exists and takes the form D = φ{d m I u (d l E(u) d t E (u))}, (1.3) where E(u) = u k u l / u 2 is the 2 2 matrix representing orthogonal projection along the direction of the velocity vector and E (u) = I E(u) its orthogonal complement, and φ = φ(x) is the porosity of the rock; d m, d l and d t are the molecular diffusion coefficient, the longitudinal and transverse dispersion coefficients respectively. The conservation of mass of the ith component in the mixture implies the following equation 8: φ c i t φz ic i p t (c i u) z i c i u p (D c i ) z i D c i p = ĉ i q, (1.4) where q = q(x, t) is the external volumetric flow rate, and q its positive part, and ĉ i is the concentration of the ith component in the external flow; ĉ i must be specified at injection points(i.e., q > ) and ĉ i = c i at production points. Since we assume the fluid to occupy the void space in the rock, n c j (x, t) = j=1 n ĉ j (x, t) = 1. (1.5) j=1

3 The effect of numerical integration in Galerkin methods 19 Add the n equations of (1.4) and use (1.5) to obtain the pressure equation n φ z j c j p u = q. (1.6) t j=1 The numerical method that we shall analyze below can be applied to the n component model, however, for clarity of presentation we shall confine ourselves to a two component displacement problem. If the components are of "slight compressibility" and if we also consider only molecular diffusion, so that D = φ(x)d m I, then a set of equations 8 modeling the pressure p and the concentration c subject to the initial and the no flow boundary conditions is given by d(c)p t u = d(c)p t (a(c) p) = q, (x, t) Ω (, T, φc t b(c)p t u c u (D c) = (ĉ c)q, (x, t) Ω (, T, u ϑ = on Ω, D c ϑ = on Ω, (1.7) where p(x, ) = p (x), x Ω, c(x, ) = c (x), x Ω, c = c 1 = 1 c 2, c t = c t, p t = p t, a(c) = a(x, c) = k(x) 2 µ(c) ; d(c) = d(x, c) = φ(x) z j c j b(c) = b(x, c) = φ(x)c 1 {z 1 ϑ is the exterior normal to Ω, the boundary of Ω. j=1 2 z j c j } j=1 The effect of numerical integration in finite element method for solving elliptic equations, parabolic and hyperbolic equations has been analyzed by Raviart2, Ciarlet and Raviart3, So- Hsiang Chou and Li Qian 4, Li Qian and Wang Dao Yu5 and others. The pressure p appears in the concentration only through its velocity field and it is appropriate to choose a numerical method that approximates the velocity u directly. In this paper we shall formulate and must consider the use of the numerical quadrature scheme based on approximating the solution of the differential system (1.7). It approximates the concentration of one of the fluids c and the pressure of the mixture p by a standard Galerkin method. We shall also give some sufficient conditions on the quadrature scheme which insure that the order of convergence in the absence of numerical integration is unaltered by the effect of numerical integration.

4 2 Nguimbi Germain et al. 2 Notation and formulation of the finite element procedures The inner product on L 2 (Ω) or L 2 (Ω) 2 is denoted by (f, g) = Ω fgdx. We shall consider W m,s (Ω), H m (Ω) = W m,2 (Ω), L 2 (Ω) = H (Ω) = W,2 (Ω) and L s (Ω) = W,s (Ω) for any integer m and any number s such that 1 s, as the usual Sobolev and Lebesgue spaces on Ω respectively. The associated norms are denoted as follows: m,s = W m,s (Ω), m = H m (Ω), = L 2 (Ω) or L 2 (Ω) 2 = L (Ω). as appropriate, Let X be any of L s or Sobolev spaces. For a function f(x, t) defined on Ω, T we let f 2 L 2 (X) = T f(, t) 2 X dt, f L (X) = ess sup f(, t) X. t T We make the following assumptions on the data (i) the solution is smooth; i.e., q is smoothly distributed, (ii) the coefficients a(c), b(c), d(c), φ(x) and D are smooth, (iii) there exist uniform positive constants a, a 1, d, φ, φ 1, D, b and M such that a a(c) a 1 ; d d(c); φ φ(x) φ 1 ; D D(x); b b(c); a c a cc t a(c) q t b(c) t d(c) M, (iv) the domain has at least the regularity required for a standard elliptic Neumann problem to have H 2 (Ω)-regularity and more if the piecewise polynomial spaces used in the finite element procedures have degrees greater than one. (v) compatibility requires that (q, 1) = Ω q(x, t)dx =, t T. Let h = (h c, h p ), where h c and h p are positive. Let M h = M hc W 1, (Ω) be a piecewise polynomial space of degree at least l associated with a quasi-regular polygonalization T hc and having the following approximation and inverse hypotheses: of Ω inf z z h 1,q M 1 h l c z l1,q, z W l1,q (Ω), 1 q, (a) z h M h z h 1, Mh 1 c z h 1, (b) z h Mh 1 c z h, (c) z h 1 Mh 1 c z h, z h M h, (M is independent of h c ) (c) (2.1)

5 The effect of numerical integration in Galerkin methods 21 Let N h W 1, (Ω) be a piecewise polynomial space of degree at least k associated with another quasi-regular polygonalization T hp of Ω and having the following approximation and inverse properties: inf v v h 1,q Mh k p v k1,q, 1 q, v W k1,q (Ω) (a) v h N h v h 1, Mh 1 p v 1, (b) v h Mh 1 p v h, (c) v h 1 Mh 1 p v h, v h N h, (M is independent of h p ) (d) (2.2) The weak form of (1.7) is defined by finding a map {c, p} :, T H 1 (Ω) H 1 (Ω) such that (φc t, z) (u c, z) (D c, z) (b(c)p t, z) = ((ĉ c)q, z), z H 1 (Ω), < t T, (a) (d(c)p t, v) (a(c) p, v) = (q, v), v H 1 (Ω), < t T, (b) c() = c, (c) p() = p. (d) (2.3) When numerical integration is not used, problem (2.3) has been studied by J. Douglas JR. and J.E. Roberts 8 where optimal L 2 -estimates are obtained. Following 1, we give a general description of the corresponding formulation of (2.3) when numerical integration is present. In what follows let f be c or p as appropiate and s be l or k as appropriate and S h be M h or N h as appropriate. Let T hf be a quasi-regular polygonalization of the set Ω with elements (K f, P Kf, Σ Kf ) with diameters h f. The following assumptions shall be made (i) The family (K f, P f, Σ f ), K f T hf for all h f is a regular affine family with a single reference finite element ( ˆK f, ˆP f, ˆΣ f ). (ii) ˆP f = P s ( ˆK f ), the set of polynomials of degree less than or equal to s. (iii) The family of triangulations or quadrilateralizations h f T hf satisfies an inverse hypothesis. (iv) Each polygonalization T hf is associated with a finite-dimensional subspace S h of trial functions which is contained in H 1 (Ω) C (Ω). We now introduce a quadrature scheme over the reference set ˆK f. A typical integral ˆφ(ˆx)dˆx ˆKf is L f approximated by ˆω lf ˆφ(ˆblf ), where the points ˆb lf ˆK f and the numbers ˆω lf >, 1 l f L f l f =1

6 22 Nguimbi Germain et al. are repectively the nodes and the weights of the quadrature. Let F Kf : ˆx ˆK f x F Kf (ˆx) B Kf ˆxb Kf be the invertible affine mapping from ˆK f onto K f with the Jacobian of F Kf, det(b Kf ) >. Any two functions φ and ˆφ on K f and ˆK f are related as φ(x) = ˆφ(ˆx) for all x = F Kf (ˆx), ˆx ˆK f. The induced quadrature scheme over K f is L f φ(x)dx = det(b Kf ) ω lf,k f φ(b lf,k f ), K f ˆK fˆφ(ˆx)dˆx l f =1 with ω lf,k f det(b Kf )ˆω lf, and b lf,k f F Kf (ˆb lf ), 1 l f L f. Accordingly, we introduce the quadrature error functionals L f E Kf (φ) φ(x)dx ω lf,k f φ(b lf,k f ), (2.4) K f l f =1 L Ê( ˆφ) f ˆω lf ˆφ(ˆblf ), (2.5) ˆK fˆφ(ˆx)dˆx which are related by E Kf (φ) = det(b )Ê( ˆφ). Kf (2.6) The quadrature scheme is exact for the space of functions ˆφ, if Ê( ˆφ) = ˆφ. If the approximation for the concentration, the pressure and the Darcy velocity are denoted by C, P and U, respectively, then using these quadrature formulas, the continuous-time approximation procedure of (2.3) is given by finding a map {C, P } :, T M h N h such that (φc, z) h (U C, z) h (D C, z) h (b(c)p t, z) h = ((ĉ C)q, z) h, z M h, t T, (a) l f =1 (d(c)p t, v) h (a(c) P, v) h = (q, v) h, v N h, t T, (b) U = a(c) P, (c) (2.7) P () = p, C() = c, (d) (e) where c and p are the elliptic projections of c and p respectively, defined in (2.8) and (2.9) below, (V, W ) h = L f K f T hf l f =1 ω lf,k f (V W )(b lf,k f ); C() = C(x, ), P () = P (x, ). It is frequently valuable to decompose the anlysis of the convergence of finite element methods by passing through a projection of the solution of the differential problem into the finite element

7 The effect of numerical integration in Galerkin methods 23 space. First let c :, T M h be the projection of c given by (D (c c), z) (u (c c), z) λ(c c, z) =, z M h, t T, (2.8) where λ is chosen to be large enough to insure the coercivity of the bilinear form over H 1 (Ω). Next let p :, T N h satisfy where β assures coercivity over H 1 (Ω). (a(c) (p p), v) β(p p, v) =, v N h, t T, (2.9) Let ζ = c c and η = p p. Standard arguments in the theory of Galerkin methods for elliptic problems show that ζ h c ζ 1 M c l1 h l1 c, (a) ζ M c l1, h l c, (b) η h p η 1 M p k1 h k1 p, (c) η M p k1, h k p, (d) where M depends on bounds for lower order derivatives of c and p. Also ζ t h c ζ t 1 M{ c l1 c t l1 }h l1 c, (a) η t h p η t 1 M{ p k1 p t k1 }h k1 p, (b) (2.1) (2.11) where M depends on lower order derivatives of c, p and their first derivatives with respect to time. There exists a constant K such that p L (Ω)) c L (Ω)) p t L (Ω)) c t L (Ω)) p L 2 c (Ω)) t L (Ω)) p t L (Ω)) K. (2.12) Also for 1 r 1 l 1, 1 r 2 k 1, c L (H r 1 (Ω)) c t L (H r 1 (Ω)) c tt L (H r 1 (Ω)) p L (H r p t L (H r p tt L (H r K (2.13) 3 Lemmas We point out that the general point of view in Ciarlet1 for elliptic problems has provided a guide line for our developpement here. In what follows, the letter K will denote a positive constant, not necessarily the same at different places, and ε will denote a generic positive small constant. Also K i, 1 i 1 will denote positive constant.

8 24 Nguimbi Germain et al. Lemma Assume that, for some integer s 1, (i) ˆP f = P s ( ˆK f ), (ii) the union L f l f =1 } {ˆblf contains a P s ( ˆK f )-unisolvent subset and/or the quadrature scheme is exact for the space P 2s ( ˆK f ). Then K 1 w h w K 2 w h, w S h, (w 1, w 2 ) h K w 1 h w 2 h, w 1, w 2 S h, w 2 h = (w, w) h. Lemma Assume, g C (K f ). Then for all w 1, w 2 S h, E Kf (gw 1 w 2 ) K g L (Kf ) w 1 L 2 (Kf ) w 2 L 2 (Kf ), where E Kf ( ) is the quadrature error functional in (2.4) Lemma Assume that for some integer s 1, ˆP = P s ( ˆK f ) and that Ê( ˆφ) =, ˆφ P 2s 1 ( ˆK f ). Then there exists a constant K independent of K f T hf and h f such that for any g W s1, (K f ), q P s (K f ), q P s (K f ), E Kf (gq xi q x j ) K h s1 f,k f g W s1, (Kf ) q H s (Kf ) q H 1 (Kf ), where h f,kf = diam(k f ). Lemma Under the same hypotheses as in Lemma 3.3. Furthermore assume that there exists a number q satisfying s 1 > 2 q. Then there exists a constant K independent of K f T hf and h f such that for any g W s1,q (K f ) and any w P s (K f ), E Kf ( gw) K h s1 f,k f (meas(k f )) q g W s1,q (K f ) w H 1 (K f ).

9 The effect of numerical integration in Galerkin methods 25 4 Error Estimates Our main concern in this section is to derive the errors C c, U u and P p. Introducing bases in M h and N h, (2.7) can be written as a system of ordinary differential equations. Then the assumptions on the data guarantee the solution of this system exists and is unique for t, T. Theorem 4.1. Let u, {c, p}, {C, P, U}, { c, p}, satisfy (1.2) (2.3),(2.7),(2.8) and (2.9) respectively. Let f denote c or p as appropriate and s denote l or k as approriate. Assume that (i) ˆPf = P s ( ˆK f ), (ii) the quadrature scheme ˆ Kf Lf ˆφ(ˆx)dˆx = l f =1 exact for the space P 2s 1 ( ˆK f ), and the union ˆω lf ˆφ(ˆblf ), is exact for the space P 2s ( ˆK f ) and/or L f l f =1 {ˆb lf } contains a P s ( ˆK f )-unisolvent subset. For l 1, k 1, and h = (h p, h c ) sufficiently small, there exists a constant K such that c C L h c c C L (H 1 (Ω)) (c C) t L 2 (L p P L h p u U L 2 (p P ) t L 2 (L K {h k1 p h l1 c }. Theorem 4.2. Under the hypothesis of Theorem 4.1, we have c C L (Ω)) p P L (Ω)) K (h k1 p h l1 c )(log h 1 p Proof. of Theorem 4.1 u U L (Ω)) K (h k p h l1 c h 1 p ) log h 1 c ), Let ζ = c c, ξ = c C, then c C = ζ ξ, ξ M h and ξ() =, (A) η = p p, π = p P, then p P = η π, π N h and π() =, (B) E(w 1 w 2 ) = (w 1, w 2 ) (w 1, w 2 ) h. We first consider the pressure equation. Substract (2.7b) from (2.3b), apply (2.9), set v = π t, integrate with respect to t and use the following (a(c) π, π t ) h = 1 d 2 dt (a(c) π, π) h 1 2 (a c(c)ξ t π, π) h 1 2 (a c(c) c t π, π) h

10 26 Nguimbi Germain et al. to obtain (d(c)π t, π t ) h dτ 1 2 (a(c) π, π) h = 1 2 (a c (C)ξ t π, π) h dτ 1 2 {((a(c) a(c)) p, π t ) h E(a(C) p t π t )}dτ E(d(c) p t π t )dτ It follows from Lemma 3.1 that we have (d(c)η t, π t )dτ (d(c)π t, π t ) h dτ 1 2 (a(c) π, π) h d R 1 R 2 R 4 R 6 R 7 K ξ t π π dτ η t π t dτ µ(η, π t )dτ (a c (C) c t π, π) h dτ ((d(c) d(c)) p t, π t ) h dτ E(qπ t )dτ = 8 R i. (4.1) i=1 π t 2 dτ K π 2 (4.2) η π t dτ c t π 2 dτ Make the induction hypothesis that π K 3, for some constant K 3. Apply (2.1),(2.11) and (4.3) to obtain R 1 R 2 R 4 R 6 R 7 K 4 h 2(l1) c h 2(k1) p { π 2 ξ 2 }dτ where K 4 depends on K 3 and c t, p t, c l1, p k1 and p t k1. Integrate R 3 by parts with respect to time to see that R 3 = (a(c) p, π) h (a(c) p, π) (a(c) p t, π)dτ (a c (C)C t p, π) h dτ = ((a(c) a(c)) p, π) E(a(C) p π) (a c (c)c t p, π)dτ E(a c (C)C t p π)dτ ε C c p t π t dτ (4.3) { π t 2 ξ t 2 }dτ, (4.4) (a(c) p t, π) h dτ ((a c (C)C t a c (c)c t ) p, π)dτ E(a(C) p t π)dτ ((a(c) a(c)) p t, π)dτ = 6 Q i. i=1 Then using the following inequality ξ 2 K ξ 2 dτε ξ t 2 dτ, (4.5)

11 The effect of numerical integration in Galerkin methods 27 { Q 1 Q 4 Q 6 K ( p 1, p t 1, )( ζ ξ ) p 1, ( ζ t ξ t ) } π dτ p 1, ( ζ ξ ) π K h 2(l1) c ξ 2 ( π 2 ξ 2 )dτ ε π 2 ξ t 2 dτ K h 2(l1) c ( π 2 ξ 2 )dτ ε π 2 ξ t 2 dτ, (4.6) where K depends on the L (W 1, (Ω))-norms of p and p t. Notice that by the interpolation theory in Sobolev spaces we have where π h c denotes the M h -interpolant of c. Then c π h c c c c π h c K h l1 c, (4.7) c π h c K h l1 c, (4.8) Q 2 = E((a(C) a( c)) p π)e((a( c) a(π h c)) p π)e((a(π h c) a(c)) p π)e(a(c) p π). Apply (4.5),(4.7),(4.8), and Lemmas 3.2 and 3.3 and the inequality a Kf b Kf ( a Kf 2 ) 1/2 ( b Kf 2 ) 1/2 to obtain K f K f K f ( ) ( E Q 2 E a c ( c τξ)dτ p( ξ) π a c (π h c τ( c π h c))dτ p( c π h c) π) E((a(π h c) a(c)) p π) E(a(c) p π) K a c ( c τξ)dτ p ξ π a c (π h cτ( c π h c))dτ p c π h c π a(π h c) a(c) p π h k1 p a(c) W k1, (Ω) p H k (Kp) π H 1 (Kp) K p K hc 2(l1) h 2(k1) p ξ 2 dτ ε π 2 1 ξ t 2 dτ, (4.9) where K depends on the L (H k (Ω))-norms of p, p L and p L (Ω)). Q 3 = = E(a c (C)C t p π)dτ = E( a c (C)ξ t p π)dτ E((a c (π h c) a c (c)) c t p π)dτ E( a c (C)ξ t p π)dτ E((a c (C) a c ( c)) c t p π)dτ E(a c (C) c t p π)dτ E(a c (c) c t p π)dτ. E((a c ( c) a c (π h c)) c t p π)dτ

12 28 Nguimbi Germain et al. Similar to the estimation of Q 2, we have Q 3 K a c (C) p ξ t π dτ a cc (π h c τ( c π h c))dτ( c π h c) c t p (a(π h c) a(c)) c t p π dτ K hc 2(l1) h 2(k1) p where K depends on c t k1;. Similar to the estimation of Q 2, we have Q 5 = thus Therefore E(a(C) p t π)dτ = ( π 2 1 ξ 2 )dτ Q 5 K ξ π dτ a cc ( c τξ)dτ c t p h k1 p ε E((a(C) a( c)) p t π)dτ c π h c π dτ ξ π dτ a(c) c t W k1, (Ω) K p p H k (Kp) π H 1 (Kp) dτ ξ t 2 dτ, (4.1) E((a(π h c) a(c)) p t π)dτ K h 2(l1) c h 2(k1) p R 3 K h 2(l1) c hp 2(k1) Integrate by parts in time to see that Use Lemma 3.4 to obtain R 5 = E(d(c) p t π) R 5 K h k1 p p t H k1 π 1 K hp 2(k1) π 2 1dτ h l1 c π dτ ( π 2 1 ξ 2 )dτ ( π 2 1 ξ 2 )dτ E(a( c) a(π h c)) p t π)dτ E(a(c) p t π)dτ, h k1 p π 1 dτ. (4.11) ε π 2 1 E((d(c) p t ) t π)dτ. ξ t 2 dτ K h k1 p { p t H k1 p tt H k1} π 1 dτ ε π 2 1, where K depends on the L (H k1 (Ω))-norms of p t and p tt. Since compatibility requires that Ω q(x, t)dx =, t T, then R 8 =. We can now combine (4.2) and the above R i, 1 i 8 to obtain π t 2 dτ π 2 1 K 5 h 2(l1) c h 2(k1) p ( π 2 1 ξ 2 )dτ ε. ξ t 2 dτ, (4.12)

13 The effect of numerical integration in Galerkin methods 29 where K 5 depends, in particular, on the induction bound K 3 for π in L (Ω). We now turn to the examination of the concentration equation. Substract (2.8) from (2.3a) to obtain (φ c t, z)(u c, z)(d c, z)(b(c) p t, z) = (( c c)q, z) (φζ t, z)λ(ζ, z) (b(c)η t, z), z N h, (4.13) Note that (ζ ξ), q >, (ĉ c) (ĉ C) =, q <. (4.14) Substract (2.7a) from (4.13) and apply (4.14) to obtain (φξ t, z) h (U ξ, z) h (D ξ, z) h (b(c)π t, z) h = ((ζξ)q, z)e((ĉ C)qz) (φζ t, z)λ(ζ, z) (b(c)η t, z)((u u) c, z)((b(c) b(c)) p t, z) E(U cz) E(D c z) E(b(C) p t z)) E(φ c t z) Take z = ξ t, observe that d dt (D ξ, ξ) h = 2(D ξ, ξ t ) h and integrate in time to see that t, ξ t ) h dτ (φξ 1 d 2 dt (D ξ, ξ) h = E((Ĉ C)qξ t)dτ ((b(c) b(c)) p t, ξ t )dτ (φζ t, ξ t )dτ (U ξ, ξ t ) h dτ E(U cξ t )dτ λ(ζ, ξ t )dτ E(D c ξ t )dτ (b(c)π t, ξ t )dτ (b(c)η t, ξ t )dτ E(b(C) p t ξ t )dτ ((ζ ξ)q, ξ t )dτ ((U u) c, ξ t )dτ E(φ c t ξ t )dτ = 13 F i i=1 (4.15) It follows from Lemma 3.1 that (φξ t, ξ t ) h dτ 1 2 (D ξ, ξ) h φ we have by using (2.1) and (2.11) F 1 F 2 F 3 F 5 F 6 F 7 F 9 K U ξ ξ t dτ K 6 hc 2(l1) hp 2(k1) π t ξ t dτ ξ t 2 dτ K ξ 2, (4.16) ( ζ ξ ) ξ t dτ ( ξ 2 ξ 2 π t 2 )dτ ε ζ t ξ t dτ ξ t 2 dτ, η t ξ t dτ

14 3 Nguimbi Germain et al. where K 6 depends on U, p t, c l1, c t l1, p k1, p k1. Since compatibility requires that (q, 1) = Ω q(x, t)dx =, t T, then F 4 =. Next, use (1.2) and (2.7c), we observe that F 8 = (a(c) η c, ξ t )dτ (a(c) π c, ξ t )dτ ((a(c) a(c)) P c, ξ t )dτ. Use c = c ξ in the first term of the right-hande side, and use Green s formula for the term involving η c, and integrate by parts in time the term involving ξ t to obtain F 8 = (η, a(c) c ξ) (a(c) η ζ, ξ t )dτ (η t, a(c) c ξ)dτ (a(c) π c, ξ t )dτ (η(a(c) c) t, ξ)dτ (η, (a(c) c)ξ t )dτ ((a(c) a(c)) P cξ t )dτ. Then use (2.1b) and (2.1d) and assume that there exists K such that P L (Ω)) K 6 F 8 K η ξ η t ξ dτ π c ξ t dτ K h 2(k1) p h 2(l1) c η ξ dτ η ξ t ( ζ ξ ) P c ξ t dτ ( ξ 2 ξ 2 ) π 2 )dτ ε ξ 2 η ζ ξ t dτ ξ t 2 dτ where K depends on c l1, c l1,, p k1, p t k1, p k1, and the L (W 1, (Ω))-norm of c. Similarly we have F 1 = E(U cξ t )dτ = E(a(c) c ηξ)dτ E((a(c) a(c)) P cξ t )dτ E(a(c) π cξ t )dτ E(a(c) p cξ t )dτ Since E( ab) = E(a b), hence by Lemmas 3.2 and 3.4,(2.1b),(2.1d) and (4.5) we have F 1 K a(c) c η ξ (a(c) c) η ξ t dτ ( ζ ξ ) P c ξ t dτ K 7 hp 2(k1) hp 2(k1) a(c) c η t ξ dτ h k ph l c ξ t dτ (a(c) c) t η ξ dτ π c ξ t dτ p c ξ ( p c p t c p c t ) ξ dτ ( ξ 2 ξ 2 π 2 )dτ ε ξ 2 ξ t 2 dτ,,

15 The effect of numerical integration in Galerkin methods 31 where K 7 depends on the L (W 1, (Ω))-norm of c t and various norms of the solution of (1.7). Integrate by parts in time to sse that F 11 = E(D c ξ t )dτ = E(D c ξ) Thus by Lemma 3.3 we have F 11 K hc l1 D W l1 (Ω) c H l (Ω) ξ H 1 (Ω) K h 2(l1) c ξ 2 1dτ ε ξ 2 1, where K depends on the L (H 1 (Ω))-norms of c and c t. F 12 = E(b(C) p t ξ t )dτ = E((b(π h c) b(c)) p t ξ t )dτ Similar as in estimation of Q 2, we obtain F12 1 F12 2 F12 3 K ξ ξ t dτ E((b(C) b( c)) p t ξ t )dτ h l1 c E(D c t ξ)dτ. h l1 c D W l1 (Ω) c t H l (Ω) ξ 1 dτ E(b(c) p t ξ t )dτ = E((b( c) b(π h c)) p t ξ t )dτ 4 i=1 ξ t dτ K h 2(l1) c F i 12. ξ 2 dτ ε ξ t 2 dτ. Integrate F 4 12 by parts in time to see that So that by Lemma 3.4 F 4 12 K h k1 p p t H k1 ξ 1 Thus F 4 12 = E(b(c) p t ξ) F 12 K hp 2(k1) hc 2(l1) E((b(c) p t ) t ξ)dτ. K h k1 p ( p tt H k1 p t H k1) ξ 1 dτ K Integrate F 13 by parts in time to see that So that by Lemma 3.4 F 13 K h l1 c c t H l1 ξ 1 ξ 2 1dτ F 13 = E(φ c t ξ) ε ξ 2 1 E((φ c t ) t ξ)dτ. ξ t 2 dτ K h l1 c ( c t H l1 c tt H l1) ξ 1 dτ K. h 2(k1) p h 2(l1) c ξ 2 1dτ ξ 2 1dτ ε ξ 2 1. ε ξ 2 1,

16 32 Nguimbi Germain et al. where K depends on the L (H l1 (Ω))-norms of c t and c tt. Combine the above F i, 1 i 13 and the relation (4.16) to obtain ξ t 2 dτ ξ 2 1 K 8 hc 2(l1) h 2(k1) p ( ξ 2 1 π t 2 π 2 )dτ, (4.17) with K 8 dependent on the induction bound K 3 for π in L (Ω). Take a (K 8 1)-multiple of (4.12) and require that the ε appearing in (4.12) be sufficiently small that (K 8 1)ε 1 2 and combine (K 8 1)-multiple of (4.12) and (4.17) to obtain ( ξ t 2 π t 2 )dτ ξ 2 1 π 2 1 K 9 h 2(l1) c h 2(k1) p ( ξ 2 1 π 2 1)dτ, (4.18) with K 9 dependent on the induction bound K 3 for π in L (Ω). Thus, it follows from Gronwall s Lemma that ξ L (H 1 (Ω)) ξ t L 2 (L π L (H 1 (Ω)) π t L 2 (L K 1 (h k1 p h l1 c ), (4.19) with K 1 depending on K 3. To complete the proof, we must show that π K 3 holds for small (h p, h c ). Use (2.2c) and (4.19) to show that π Mh 1 p π K h 1 p (h k1 p h l1 c ) K (h k p h l1 c h 1 p ) Assume that (4.2) h l1 c h 1 p as h, holds, then K 3 could have been taken to be arbitrarily small and a choice can be made for K 1 that is independent of K 3. That is,if π is bounded uniformly by some constant K 3, then π tends uniformly to zero as h p, and h c tend to zero subject to (4.2). Now, since π() =, then (4.3) holds for t = ; hence for any fixed pair (h p, h c ), (4.3) holds for t T h for some T h >. The implication of (A),(B),(2.1a) and (2.1c) is that T h = T for (h p, h c ) sufficiently small; i.e. (4.3) holds for small (h p, h c ). Therefore use (4.19),(2.1a),(2.1c),(2.11) and the triangle inequality to sse that, under the constraint (4.2), h l1 c h 1 p as h, c C L h c c C L (H 1 (Ω)) (c C) t L 2 (L p P L h p u U L 2 (Ω) 2 ) (p P ) t L 2 (L K (h k1 p h l1 c ), where K depends on spatial derivatives of order not greater than l 1 of c and c t and of order not greater than k 1 of p and p t.

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18 34 Nguimbi Germain et al. 1 M.F. Wheeler, A priori L 2 -error estimates for Galerkin approximates to parabolic partial differential equations, SIAM J. Numer. Anal., 1 (1973), J. Douglas Jr., T. Dupont and R.E. Ewing, Incomplete integration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. Num. Anal., 16 (1979), Yuan Yi-rang, Time stepping along characteristics for the finite element approximation of compressible miscible displacement in porous media, Math. Numer. Sinica, 14 (1992), no. 4, Received: October 1, 216; Published: January 1, 217