Research Article Discrete Mixed Petrov-Galerkin Finite Element Method for a Fourth-Order Two-Point Boundary Value Problem

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1 International Journal of Matematics and Matematical Sciences Volume 2012, Article ID , 18 pages doi: /2012/ Researc Article Discrete Mixed Petrov-Galerkin Finite Element Metod for a Fourt-Order Two-Point Boundary Value Problem L. Jones Tarcius Doss 1 and A. P. Nandini 2 1 Department of Matematics, Anna University Cennai, CEG Campus, Cennai , India 2 Department of Matematics, M.N.M Jain Engineering College, Toraipakkam, Cennai , India Correspondence sould be addressed to L. Jones Tarcius Doss, jones@annauniv.edu Received 20 July 2011; Revised 24 November 2011; Accepted 25 November 2011 Academic Editor: Attila Gilányi Copyrigt q 2012 L. J. T. Doss and A. P. Nandini. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. A quadrature-based mixed Petrov-Galerkin finite element metod is applied to a fourt-order linear ordinary differential equation. After employing a splitting tecnique, a cubic spline trial space and a piecewise linear test space are considered in te metod. Te integrals are ten replaced by te Gauss quadrature rule in te formulation itself. Optimal order apriorierror estimates are obtained witout any restriction on te mes. 1. Introduction In tis paper, we develop a quadrature-based Petrov-Galerkin mixed finite element metod for te following fourt-order boundary value problem: [ d 2 a x d2 u b x u f x, x I 0, 1, 1.1 dx 2 dx 2 subject to te boundary conditions u 0 0, u 1 0; u 0 0, u 1 0, 1.2 were a x / 0, x I. Letα x 1/a x. We, ereafter, suppress te dependency of te independent variable x on te functions α x, b x, andf x. Terefore, we write α, b, and f instead of tese functions. Let us define te splitting of te above fourt-order equation as follows.

2 2 International Journal of Matematics and Matematical Sciences Set u αv, x I. 1.3 Ten te differential equation 1.1 wit te boundary conditions 1.2 can be written as a coupled system of equations as follows: u αv, x I, wit u 0 u 1 0, 1.4 v bu f, x I, wit v 0 v In tis paper, te error analysis will take place in te usual Sobolev space Wp m I defined on te domain I 0, 1 wit H m I denoting W m 2 I. Te Sobolev norms are given below. For an open interval E and a non negative integer m, ( m 1/p v W m p E v i p, if 1 p<, L i 0 p E 1.6 max v i, if p. L 1 i n E We suppress te dependence of te norms on I wen E I. Furter, H m 0 I denotes te function space { φ H m I : φ 0 φ 1 0 } Continuous and Discrete H 1 -Galerkin Formulation Given n>1, let Π n :0 x 0 <x 1 < <x n be an arbitrary partition of 0, 1 wit te property tat 0asn, were max 1 k n k and k x k x k 1, k 1,...,n.Let u, v represent te L 2 inner product, and let u, v represent te discrete inner product of any two functions u, v L 2 I and be defined as follows: u, v uv dx, u, v Q uv, 2.2 were Q is te fourt-order Gaussian quadrature rule: ( 1 n [ Q g : k g xk,1 g x k,2. 2 i 1 2.3

3 International Journal of Matematics and Matematical Sciences 3 Here, x k,i x k 1 ξ i k,i 1, 2, are te two Gaussian points in te subinterval x k 1,x k wit ξ 1 1/2 1 1/ 3, ξ 2 1 ξ 1. Let us now consider te following cubic spline space as trial space: S,3 { } ϕ C 2 I : ϕ Ik P 3 I k,k 1, 2,...,n, 2.4 were P r I k is te space of polynomials of degree r defined over te kt subinterval I k x k 1,x k. Te corresponding space wit zero Diriclet boundary condition is denoted by 0 S,3 { ϕ S,3 : ϕ 0 ϕ 1 0 }. 2.5 Furter, let us consider te following piecewise linear space S,1 { ϕ C I : ϕ Ik P 1 I k, k 1, 2,...,n } 2.6 as te test space Weak Formulation Te weak formulation corresponding to te split equations 1.4 and 1.5 is defined, respectively, as follows. Find {u, v} H 2 0 I suc tat ( u,φ ( αv, φ, φ H 2 0, 1, 2.7 ( v bu, φ ( f, φ, φ H 2 0, Te Petrov-Galerkin Formulation Te Petrov-Galerkin formulation corresponding to te above weak formulation 2.7 and 2.8 is defined, respectively, as follows. Find {u,v } 0 S,3 suc tat ( u,φ ( αv,φ, φ S,1, (v bu,φ ( f, φ, φ S, Te integrals in te above Petrov-Galerkin formulation are not evaluated exactly at te implementation level. We, terefore, define te following discrete Petrov-Galerkin procedure in wic te integrals are replaced by te Gaussian quadrature in te sceme as follows.

4 4 International Journal of Matematics and Matematical Sciences 2.3. Discrete Petrov-Galerkin Formulation Te discrete Petrov-Galerkin formulation corresponding to 2.7 and 2.8 is defined, respectively, as follows. Find {u,v } 0 S,3 suc tat u,φ αv,φ, φ S,1, 2.10 v bu,φ f, φ, φ S, Te approximate solutions u and v witout any conditions on boundary points are expressed as a linear combination of te B-splines as follows: n 1 u x γ j B j x, j 1 n 1 v x δ j B j x, j were te jt basis B j x of te cubic B-splines space S,3 for j 1, 0, 1, 2,...,n,n 1isgiven below: 0, if x x j 2, 1 ( 3, x xj 2 if xj 2 x x 6 3 j 1, 1 ( 3 3 2( ( 2 ( 3 x x 6 B j x 3 j 1 3 x xj 1 3 x xj 1, if x j 1 x x j, 1 ( 3 3 2( x 6 3 j 1 x 3 ( x j 1 x 2 ( 3 xj 1 x 3, if x j x x j 1, 1 ( xj 2 x 3, if xj 1 x x 6 3 j 2, 0, if x x j For j 1, 0andj n, n 1, te basis functions are defined as in te above form, after extending te partition by introducing fictitious nodal points x 3,x 2,x 1 on te left-and side and x n 1,x n 2,x n 3 on te rigt-and side, respectively. Furter, te it basis φ i x of te piecewise linear at splines space S,1 for i 0, 1, 2,...,nis given below: 0, if x x i 1, 1 x x i 1, if x i 1 x x i, φ i x 1 x i 1 x, if x i x x i 1, 0, ifx x i

5 International Journal of Matematics and Matematical Sciences 5 In a similar manner, for i 0andi n, te basis functions are defined as in te above form, after extending te partition by introducing fictitious nodal point x 1 on te leftand side and x n 1 on te rigt-and side, respectively. Te mixed discrete Petrov-Galerkin metod for 2.10 and 2.11 witout assuming boundary conditions in te trial space is given as follows: n 1 γ j j 1 n 1 γ j bbj,φ i j 1 n 1 B j,φ i j 1 n 1 δ j αbj,φ i 0, i 0, 1, 2,...,n, δ j j 1 B j,φ i f, φ i, i 0, 1, 2,...,n, 2.15 wit te corresponding equations: n 1 γ j B j 0 0, j 1 n 1 δ j B j 0 0, j 1 n 1 γ j B j 1 0, j 1 n 1 δ j B j 0 0, j referring to te zero-boundary conditions: u 0 0, u 1 0, v 0 0, v Te above set of equations can be written as a set of 2n 6 equations in 2n 6 unknowns. Here, we study te effect of quadrature rule in te error analysis. Since we compute te approximations for te solution u x as well as for its second derivative v x wit integrals replaced by Gaussian quadrature rule in te formulation, tis work may be considered as a quadrature-based mixed Petrov-Galerkin metod. 3. Overview of Discrete Petrov-Galerkin Metod Here, te integrals are replaced by composite two-point Gauss rule. Terefore, te resulting metod may be described as a qualocation approximation, tat is, a quadrature-based modification of te collocation metod. Furter, it may be considered as a Petrov-Galerkin metod wit a quadrature rule because te test space and trial space are different. Hence, it may be referred to as discrete Petrov-Galerkin metod. One practical advantage of tis procedure over te ortogonal spline collocation metod described in Douglas Jr. and Dupont 1, 2 is tat for a given partition tere are only alf te number of unknowns, and terefore it reduces te size of te matrix. Te qualocation metod was first introduced and analysed by Sloan 3 for boundary integral equation on smoot curves. Later on Sloan et al. 4 extended tis metod to a class of linear second-order two-point boundary value problems and derived optimal error estimates witout quasi-uniformity assumption on te finite element mes. Ten, Jones Doss and Pani 5 discussed te qualocation metod for a second-order semilinear two-point boundary

6 6 International Journal of Matematics and Matematical Sciences value problem. Furter, Pani 6 expanded its scope by adapting te analysis to a semilinear parabolic initial and boundary value problem in a single space variable. Jones Doss and Pani 7 extended tis metod to te free boundary problem, tat is, one-dimensional singlepase Stefan problem for wic part of te boundary as to be found out along wit te solution process. A quadrature-based Petrov-Galerkin metod applied to iger dimensional boundary value problems is studied in Bialecki et al. 8, 9 and Ganes and Mustapa 10. Te main idea of tis paper is tat a quadrature based approximation for a fourt order problem is analyzed in mixed Galerkin setting. Te organization of tis paper is as follows. In previous Sections 1 and 2, te problem is introduced; te weak and te Galerkin formulations are defined. Overview of discrete Petrov-Galerkin metod is discussed in Section 3. Preliminaries required for our analysis are mentioned in Section 4. Error analysis is carried over in Section 5. Trougout tis paper C is a generic positive constant, wose dependence on te smootness of te exact solution can be easily determined from te proofs. 4. Preliminaries We assume tat α and b are suc tat α, b C 4( I, 4.1 were I 0, 1. We assume tat te problem consisting of te coupled equations 1.4 and 1.5 is uniquely solvable for a given sufficiently smoot function f x. It can be proved tat te quadrature rule in 2.3 as an error bound of te form ( ( n E g Q g g C 4 k i 1 g 4. L1 I k 4.2 Tis follows from Peano s kernel teorem see 11. Te following inequality is frequently used in our analysis. If v W m p E wit p 1,, ten tere exists a positive constant C depending only on m suc tat, for any δ satisfying 0 <δ E 1, v W i p E [δ C m i v W m p E δ i v Lp E, 0 i m 1, 4.3 were E denotes te lengt of E. For a detailed proof, one may refer to appendix of Sloan et al. 4 or Capter 4 of Adams 12. Let us use te following notation: Lv : v. 4.4 Te adjoint operator L wit corresponding adjoint boundary condition is defined as follows: L φ φ, φ 0 φ

7 International Journal of Matematics and Matematical Sciences 7 Since L is a self-adjoint operator, we mention below te regularity of L equal to L in te q norm. We make a stronger assumption as in Sloan et al. 4 tat for arbitrary q 1,, tere exists a positive constant C suc tat L u Lq I C u W 2 q I. 4.6 We ave te following inequality due to te Sobolev embedding teorem; te proof of wic can be found in page 97, Adams 12, φ L I k φ W 1 p I k ; 1 p,φ W 1 p I k Convergence Analysis Hereafter trougout tis section, for p and q wit 1 p, q, sandp 1 q 1 1, we use te following notations: v 0,p v Lp, v s,p v W s p, v s,p,k v W s p I k. 5.1 Let us denote te error between u and u by ε and te error between v and v by e, respectively, tat is, ε u u and e v v.using 2.11 and 1.5, we obtain te following error equations: e,φ v v,φ v,φ f bu,φ b u u,φ bε,φ, 5.2 and terefore we get e,φ bε,φ, φ S, Furter, using 2.10 and 1.4, ε,φ u u,φ α v v,φ αe,φ, 5.4 and terefore we ave ε,φ αe,φ, φ S, Te following lemma gives estimates for te error in te quadrature rule for te term e χ and ε χ for χ S,1. Tese estimates are required for our error analysis later. Te proof of te lemma is similar to te proof of Lemma 4.2 of Sloan et al. 4.

8 8 International Journal of Matematics and Matematical Sciences Lemma 5.1. For all χ S,1 andsufficiently small, a E e χ C 4 v 6,p χ 1,q, b E e χ C 3 v 6,p χ 0,q, c E ε χ C 4 u 6,p χ 1,q, d E ε χ C 3 u 6,p χ 0,q. Te following result gives estimate for ε x, were x is any arbitrary point in I. Tis estimate is crucial for our error analysis. Lemma 5.2. Let u be te weak solution of 1.4 defined troug 2.7. Furter, let u be te corresponding discrete Petrov-Galerkin solution defined troug Ten, te error ε u u satisfies ε x C [ 2 ε 2,p 4 u 6,p e 1,p, 5.6 were x is an arbitrary point in 0, 1. Proof. For a given x 0, 1, letφ be an element of L p I C I satisfying te following auxiliary problem: Φ 0, x I {x}, Φ 0 Φ 1 0, Φ x Φ x Te above problem as a solution. For example, Φ x { x 1 x, 0 x x, x x 1, x x satisfies te above differential equation, te boundary conditions, and te jump condition. Let us define Ψ as follows: Ψ x { Φ, x I {x}, 0, at x x. 5.9

9 International Journal of Matematics and Matematical Sciences 9 Ten, Ψ 0 a.e. on I. We first multiply ε wit Ψ and ten integrate over I. On applying integration by parts, using te fact tat ε 0 ε 1 0 and te jump condition for Φ,we obtain 0 ε, Ψ x 0 ε Ψ 1 x ε Ψ x 0 ε Φ 1 x ε Φ [ ε Φ x x 0 ε Φ [ ε Φ 1 1 x ε Φ ε x [ Φ x Φ x 0 x ε x x 0 ε Φ 1 x ε Φ. x 0 ε Φ 1 ε Φ x 5.10 Applying integration by parts once again, using boundary condition for Φ and te continuity of Φ, weobtain 0 ε x { [ε x 1 } Φ x 0 ε Φ [ ε Φ 1 x ε Φ ε x (ε, Φ, x tat is, ε x ε, Φ. LetΦ be te linear interpolant of Φ. Ten, we ave ( ε x ε, Φ Φ ( ε x ε, Φ Φ ( ε, Φ ( E ε Φ ε, Φ ε, Φ T 1 T 2 T 3. ε, Φ 5.12 We know tat Φ 1,q Φ Φ 1,q Φ 1,q C Φ 2,q Φ 2,q C Φ 2,q We now compute te estimates for te terms T 1, T 2,andT 3 as follows: ( T 1 ε ε, Φ Φ Φ Φ 0,q C 2 ε 2,p Φ 2,q. 0,p 5.14 Using Lemma 5.1 c and 5.13,weobtain T 2 E (ε Φ C 4 u 6,p Φ 2,q Using 5.5, 2.3, and te Sobolev embedding teorem 4.7 locally on I k for bot e 0,,k and Φ 0,,k, we ave T 3 ε, Φ αe, Φ C n k n 2 e 0,,k Φ 0,,k C k 1 k 1 k 2 e 1,p,k Φ 1,q,k. 5.16

10 10 International Journal of Matematics and Matematical Sciences Using Hölder s inequality for sums and 5.13, we ave T 3 C e 1,p Φ 1,q C e 1,p Φ 2,q For Φ satisfying te auxiliary problem, it is easy to verify tat Φ 2,q K, were K is a constant not depending on. Using T 1, T 2,andT 3 in 5.12, we ave ε x C [ 2 ε 2,p 4 u 6,p e 1,p Tis completes te proof. In te following lemma, we initially compute te error v v in terms of u u, and ten later on we establis an optimal estimate of error v v independent of u u. Lemma 5.3. Let u and v be te weak solutions of te coupled equations 1.4 and 1.5 defined troug 2.7 and 2.8, respectively. Furter, let u and v be te corresponding discrete Petrov- Galerkin solutions defined troug 2.10 and 2.11, respectively. Ten te estimates of te errors e v v in L p, W 1 p, and W 2 p norms are given as follows: e 0,p C [ 4 v 6,p 5 u 6,p 3 ε 2,p, e 1,p C [ 3 v 6,p 4 u 6,p 2 ε 2,p, e 2,p C [ 2 v 6,p 4 u 6,p 2 ε 2,p Proof. Let η be an arbitrary element of L q,andletφ W 2 q be te solution of te auxiliary problem L φ η, φ 0 φ We now ave ( e,η ( e,l φ ( Le,φ ( ( e,φ φ e,φ ( ( e,φ φ e,φ e,φ e,φ ( ( e,φ φ E ( e,η ( e,φ φ T 4 T 5 T 6, e φ ( E e φ e,φ, e,φ 5.21 were φ S,1 is te linear interpolant of φ.

11 International Journal of Matematics and Matematical Sciences 11 We know tat φ 1,q φ φ 1,q φ 1,q C φ 2,q φ 2,q C φ 2,q We sall compute te estimates for te terms T 4, T 5,andT 6 as follows: T 5 ( T 4 e e,φ φ φ φ0,q C 2 e φ2,q 2,p, 0,p E (e φ C 4 v 6,p φ 1,q C4 v 6,p φ 2,q by Lemma 5.1 a, Using 5.3, 2.3, and te Sobolev embedding teorem 4.7 locally on I k for φ 0,,k,we ave T 6 e,φ bε,φ n C k 1 2 ε n φ0,,k 0,,k C k k 1 k 2 ε 0,,k φ 1,q,k Using Hölder s inequality for sums, Lemma 5.2, and 5.22, weobtain [ [ T 6 C 2 ε 2,p 4 u 6,p e φ1,q 1,p C 3 ε 2,p 5 u 6,p 2 e φ2,q 1,p Substituting T 4, T 5,andT 6 in 5.21, we ave ( e,η C [ 2 e 2,p 4 v 6,p 3 ε 2,p 5 u 6,p 2 e 1,p φ 2,q Using 4.6 and te regularity of te auxiliary problem, we ave φ 2,q C η 0,q. Since η L q is arbitrary, we ave e 0,p C ( 2 e 2,p 3 ε 2,p 4 v 6,p 5 u 6,p We now estimate e via a projection argument. Let P be te ortogonal projection onto S,1 wit respect to L 2 inner product defined by (v P v,ψ 0, ψ S, Te domain of P may be taken to be L 1. From Crouzeix and Tomée 13 and de Boor 14, it is seen tat te L 2 projection is stable. Tus, P v 0,p C v 0,p. 5.29

12 12 International Journal of Matematics and Matematical Sciences Ten te error e can be interpreted in terms of te error of te above projection: e v v 0,p v P v 0,p P v v 0,p. 0,p 5.30 From te stability property 5.29, te error in te projection follows as in de Boor 14, tat is, v P v 0,p C 2 v 2,p C 2 v 4,p Ten te remaining task is to compute te estimate of P v v 0,p. For ψ S,1, ( ( P v v,ψ (P v v,ψ ( P v v v v,ψ ( v v,ψ v v,ψ using 5.28, ( ( ( P v v,ψ e,ψ e,ψ e,ψ E (e ψ e,ψ, ( ( P v v,ψ E e ψ e,ψ T 7 T 8. e,ψ 5.32 We sall compute te estimates for te terms T 7 and T 8 T 7 E (e ψ C 3 v 6,p ψ 0,q 5.33 by Lemma 5.1 b. Following te steps of computation involved in te term T 6, we obtain te estimate of T 8 as T 8 e,ψ C [ 2 ε 2,p 4 u 6,p e ψ0,q 1,p, 5.34 were we ave used te inverse inequality ψ 1,q,k 1 k ψ 0,q,k locally. Using T 7 and T 8 in 5.32, weget ( [ P v v,ψ C 3 v 6,p 2 ε 2,p 4 u 6,p e ψ0,q 1,p We now sow te above inequality for η L q to obtain P v v 0,p.

13 International Journal of Matematics and Matematical Sciences 13 Now let η be an arbitrary element of L q. Ten since v S,1, it follows from te definition of P η, 5.35,and 5.29 wit p replaced by q,tat ( 0 P v v,η P η, (P ( [ v v,η P v v,p η C 3 v 6,p 2 ε 2,p 4 u 6,p e P 1,p η 0,q [ C 3 v 6,p 2 ε 2,p 4 u 6,p e 1,p η 0,q, P v v C [ 3 v 6,p 2 ε 2,p 4 u 6,p e 1,p. 0,p 5.36 Now, from 5.30, 5.31, and 5.36, we conclude tat e C 2 v 4,p C [ 3 v 6,p 2 ε 2,p 4 u 6,p e 1,p 0,p C [ 2 v 6,p 2 ε 2,p 4 u 6,p e 1,p Now, using te fact e 2,p e 1,p e and te above estimate, we ave 0,p e 2,p e 1,p C [ 2 v 6,p 2 ε 2,p 4 u 6,p e 1,p C [ e 1,p 2 v 6,p 2 ε 2,p 4 u 6,p C [ e 1,p 2 v 6,p 2 ε 2,p 4 u 6,p Now using 4.3 wit m 2andi 1, we ave e 1,p C ( 1 e 0,p e 2,p Substituting 5.39 in te above expression, we obtain e 2,p C [( 1 e 0,p e 2,p 2 v 6,p 2 ε 2,p 4 u 6,p For sufficiently small, we ave e 2,p C [ 1 e 0,p 2 v 6,p 2 ε 2,p 4 u 6,p Using 5.41 in 5.27, e 0,p C [ 2( 1 e 0,p 2 v 6,p 2 ε 2,p 4 u 6,p 4 v 6,p 5 u 6,p 3 ε 2,p. 5.42

14 14 International Journal of Matematics and Matematical Sciences For sufficiently small,weget e 0,p C [ 4 v 6,p 5 u 6,p 3 ε 2,p Using 5.43 in 5.41, we ave e 2,p C [ 1( 4 v 6,p 5 u 6,p 3 ε 2,p 2 v 6,p 2 ε 2,p 4 u 6,p C [ 2 v 6,p 2 ε 2,p 4 u 6,p Using 5.43 and 5.44 in 5.39, we ave e 1,p C [ 1( 4 v 6,p 5 u 6,p 3 ε 2,p ( 2 v 6,p 2 ε 2,p 4 u 6,p C [ 3 v 6,p 4 u 6,p 2 ε 2,p Equations 5.43, 5.44,and 5.45 give te required result. We now compute te error estimate of ε in L p, W 1 p, and W 2 p norms as as been done in te previous case. Lemma 5.4. Let u and v be te weak solutions of te coupled equations 1.4 and 1.5 defined troug 2.7 and 2.8, respectively. Furter, let u and v be te corresponding discrete Petrov- Galerkin solutions defined troug 2.10 and 2.11, respectively. Ten te estimates of te errors ε u u in L p,w 1 p and W 2 p norms are given as follows: ε 0,p C [ 4 u 6,p e 1,p, ε 1,p C [ 3 u 6,p e 1,p, ε 2,p C [ 2 u 6,p e 1,p Proof. Let ρ be an arbitrary element of L q,andletφ Wq 2 auxiliary problem be te unique solution of te L φ ρ, φ 0 φ Ten we ave ( ε,ρ ( ε,l φ ( Lε,φ (ε,φ ( ( ε,φ φ ε,φ ε,φ ε,φ, 5.48

15 International Journal of Matematics and Matematical Sciences 15 were φ S,1 is a linear interpolant of φ, ( ε, ρ ( ( ε,φ φ E ε φ ε,φ T 9 T 10 T Following te steps involved in te computation of T 4 and T 5, we obtain te estimates of T 9 and T 10 as follows: T 9 C 2 ε 2,p φ 2,q, T 10 C 4 u 6,p φ 2,q, 5.50 by Lemma 5.1 c and Using 5.5 and 2.3 first, ten te Sobolev embedding teorem 4.7 locally on I k for φ 0,,k and e 0,,k to estimate T 11, we ave T 11 ε,φ k 1 αe,φ n k C 2 e φ1,q,k 1,p,k. n C k 1 2 e n φ0,,k 0,,k C k k 1 k 2 e 0,,k φ 1,q,k 5.51 Furter, using Hölder s inequality for sums and 5.22, weobtain T 11 C e 1,p φ 1,q C e 1,p φ 2,q Substituting te estimates T 9, T 10,andT 11 in 5.49,weobtain ( ε, ρ C [ 2 ε 2,p 4 u 6,p e 1,p φ 2,q Using 4.6 and regularity of te auxiliary problem, we ave φ 2,q C ρ o,q. Since ρ L q is arbitrary, we ave ε 0,p C [ 2 ε 2,p 4 u 6,p e 1,p Te estimate of ε 0,p can be obtained troug a projection argument as mentioned in Lemma 5.3 as ε C [ 2 u 6,p e 1,p, ,p

16 16 International Journal of Matematics and Matematical Sciences were we ave used Lemma 5.1 d. In a similar manner we can compute te estimates for ε 0,p, ε 1,p and ε 2,p as ε 0,p C [ 4 u 6,p e 1,p, ε 1,p C [ 3 u 6,p e 1,p, ε 2,p C [ 2 u 6,p e 1,p Using all te estimates from Lemmas 5.3 and 5.4, we ave te following main error estimates. Teorem 5.5. Assume tat u and v satisfy 1.4 and 1.5, respectively, wit 4.1. Assume also tat u Wp 6 and v Wp,werep 6 1,. Ten 2.10 and 2.11 ave unique solutions u 0 S,3 and v 0 S,3, respectively, and for sufficiently small, one as u u i,p C 4 i[ u 6,p v 6,p, v v i,p C 4 i[ u 6,p v 6,p, i 0, 1, Proof. Assume temporarily tat solutions u and v of 2.10 and 2.11, respectively, exist. Using 5.46 in 5.45, weobtain e 1,p C [ 3 v 6,p 4 u 6,p 2( 2 u 6,p e 1,p For sufficiently small, we ave e 1,p C ( 3 v 6,p 4 u 6,p An application of te above in 5.46,weget ε 2,p C [ 2 u 6,p 3 v 6,p Apply 5.59 in 5.56 to ave ε 0,p C [ 4 u 6,p 4 v 6,p Use 5.60 in 5.43 to get e 0,p C [ 4 v 6,p 5 u 6,p. 5.62

17 International Journal of Matematics and Matematical Sciences 17 Using 5.60 in 5.44, weobtain e 2,p C [ 2 v 6,p 4 u 6,p Using 5.61 and 5.60 in 5.39 wit e replaced by ε, we ave ε 1,p C [ 3 u 6,p 3 v 6,p Te required result can be obtained from estimates 5.59 to So far we ave assumed temporarily tat solutions u and v exist. We now discuss te existence and uniqueness of discrete Petrov-Galerkin approximation. Since te matrix corresponding to 2.10 and 2.11 wit zero boundary conditions for u and v is square, existence of u 0 S,3 and v 0 S,3 for any f C 0 I will follow from uniqueness, tat is, from te property tat te corresponding omogeneous equations ave only trivial solutions. Suppose tat u and v corresponding to u and v satisfy u αv,χ 0, v bu,χ 0, χ S, It follows from 5.61 and 5.62 wit u replaced by 0 and eventually v 0 tat, for sufficiently small, u 0,p 0, v 0,p 0, 5.66 and ence u 0andv 0. Tus, uniqueness is proved, and ence existence follows from uniqueness. References 1 J. Douglas, Jr. and T. Dupont, A finite element collocation metod for quasilinear parabolic equations, Matematics of Computation, vol. 27, pp , J. Douglas, Jr. and T. Dupont, Collocation Metods for Parabolic Equations in a Single Space Variable, vol. 385 of Lecture Notes in Matematics, Springer, Berlin, Germany, I. H. Sloan, A quadrature-based approac to improving te collocation metod, Numerisce Matematik, vol. 54, no. 1, pp , I. H. Sloan, D. Tran, and G. Fairweater, A fourt-order cubic spline metod for linear second-order two-point boundary value problems, IMA Journal of Numerical Analysis, vol. 13, no. 4, pp , L. Jones Doss and A. K. Pani, A qualocation metod for a semilinear second-order two-point boundary value problem, in Functional Analysis wit Current Applications in Science, Tecnology and Industry, M. Brokate and A. H. Siddiqi, Eds., vol. 377 of Pitman Researc Notes in Matematics, pp , Addison Wesley Longman, Harlow, UK, A. K. Pani, A qualocation metod for parabolic partial differential equations, IMA Journal of Numerical Analysis, vol. 19, no. 3, pp , L. Jones Doss and A. K. Pani, A qualocation metod for a unidimensional single pase semilinear Stefan problem, IMA Journal of Numerical Analysis, vol. 25, no. 1, pp , 2005.

18 18 International Journal of Matematics and Matematical Sciences 8 B. Bialecki, M. Ganes, and K. Mustapa, A Petrov-Galerkin metod wit quadrature for elliptic boundary value problems, IMA Journal of Numerical Analysis, vol. 24, no. 1, pp , B. Bialecki, M. Ganes, and K. Mustapa, An ADI Petrov-Galerkin metod wit quadrature for parabolic problems, Numerical Metods for Partial Differential Equations, vol. 25, no. 5, pp , M. Ganes and K. Mustapa, A Crank-Nicolson and ADI Galerkin metod wit quadrature for yperbolic problems, Numerical Metods for Partial Differential Equations, vol. 21, no. 1, pp , P. J. Davis and P. Rabinowitz, Metods of Numerical Integration, Academic Press, New York, NY, USA, 2nd edition, R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Matematics, Academic Press, New York, NY, USA, M. Crouzeix and V. Tomée, Te stability in L p and W 1 p of te L 2 -projection onto finite element function spaces, Matematics of Computation, vol. 48, no. 178, pp , C. de Boor, A bound on te L -norm of L 2 -approximation by splines in terms of a global mes ratio, Matematics of Computation, vol. 30, no. 136, pp , 1976.

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