International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 962070 | https://doi.org/10.1155/2012/962070

L. Jones Tarcius Doss, A. P. Nandini, "Discrete Mixed Petrov-Galerkin Finite Element Method for a Fourth-Order Two-Point Boundary Value Problem", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 962070, 18 pages, 2012. https://doi.org/10.1155/2012/962070

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

Academic Editor: Attila Gilányi
Received20 Jul 2011
Revised24 Nov 2011
Accepted25 Nov 2011
Published13 Feb 2012

Abstract

A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal order a priori error estimates are obtained without any restriction on the mesh.

1. Introduction

In this paper, we develop a quadrature-based Petrov-Galerkin mixed finite element method for the following fourth-order boundary value problem: 𝑑2𝑑𝑥2𝑑𝑎(𝑥)2𝑢𝑑𝑥2+𝑏(𝑥)𝑢=𝑓(𝑥),𝑥𝐼=(0,1),(1.1) subject to the boundary conditions 𝑢(0)=0,𝑢(1)=0;𝑢(0)=0,𝑢(1)=0,(1.2) where 𝑎(𝑥)0,𝑥𝐼. Let 𝛼(𝑥)=1/𝑎(𝑥). We, hereafter, suppress the dependency of the independent variable 𝑥 on the functions 𝛼(𝑥), 𝑏(𝑥), and 𝑓(𝑥). Therefore, we write 𝛼,𝑏, and 𝑓 instead of these functions.

Let us define the splitting of the above fourth-order equation as follows.

Set 𝑢=𝛼𝑣,𝑥𝐼.(1.3) Then the differential equation (1.1) with the boundary conditions (1.2) can be written as a coupled system of equations as follows: 𝑢𝑣=𝛼𝑣,𝑥𝐼,with𝑢(0)=𝑢(1)=0,(1.4)+𝑏𝑢=𝑓,𝑥𝐼,with𝑣(0)=𝑣(1)=0.(1.5) In this paper, the error analysis will take place in the usual Sobolev space 𝑊𝑚𝑝(𝐼) defined on the domain 𝐼=(0,1) with 𝐻𝑚(𝐼) denoting 𝑊𝑚2(𝐼). The Sobolev norms are given below. For an open interval 𝐸 and a non negative integer 𝑚, 𝑣𝑊𝑚𝑝(𝐸)=𝑚𝑖=0𝑣(𝑖)𝑝𝐿𝑝(𝐸)1/𝑝,if1𝑝<,=max1𝑖𝑛𝑣(𝑖)𝐿(𝐸),if𝑝=.(1.6) We suppress the dependence of the norms on 𝐼 when 𝐸=𝐼. Further, 𝐻𝑚0(𝐼) denotes the function space {𝜙𝐻𝑚(𝐼)𝜙(0)=𝜙(1)=0}.(1.7)

2. Continuous and Discrete 𝐻1-Galerkin Formulation

Given 𝑛>1, let Π𝑛0=𝑥0<𝑥1<<𝑥𝑛=1(2.1) be an arbitrary partition of [0,1] with the property that 0 as 𝑛, where =max1𝑘𝑛𝑘 and 𝑘=𝑥𝑘𝑥𝑘1,𝑘=1,,𝑛. Let (𝑢,𝑣) represent the 𝐿2 inner product, and let 𝑢,𝑣 represent the discrete inner product of any two functions 𝑢,𝑣𝐿2(𝐼) and be defined as follows: (𝑢,𝑣)=𝑢𝑣𝑑𝑥,𝑢,𝑣=𝒬(𝑢𝑣),(2.2) where 𝒬 is the fourth-order Gaussian quadrature rule: 𝒬1(𝑔)=2𝑛𝑖=1𝑘𝑔𝑥𝑘,1𝑥+𝑔𝑘,2.(2.3) Here, 𝑥𝑘,𝑖=𝑥𝑘1+𝜉𝑖𝑘,𝑖=1,2, are the two Gaussian points in the subinterval [𝑥𝑘1,𝑥𝑘] with 𝜉1=(1/2)(11/3), 𝜉2=1𝜉1.

Let us now consider the following cubic spline space as trial space: 𝑆,3=𝜑𝐶2(𝐼)𝜑|𝐼𝑘𝑃3𝐼𝑘,𝑘=1,2,,𝑛,(2.4) where 𝑃𝑟(𝐼𝑘) is the space of polynomials of degree 𝑟 defined over the 𝑘th subinterval 𝐼𝑘=[𝑥𝑘1,𝑥𝑘].

The corresponding space with zero Dirichlet boundary condition is denoted by 0𝑆,3=𝜑𝑆,3𝜑(0)=𝜑(1)=0.(2.5) Further, let us consider the following piecewise linear space 𝑆,1=𝜑𝐶(𝐼)𝜑|𝐼𝑘𝑃1𝐼𝑘,𝑘=1,2,,𝑛(2.6) as the test space.

2.1. Weak Formulation

The weak formulation corresponding to the split equations (1.4) and (1.5) is defined, respectively, as follows.

Find {𝑢,𝑣}𝐻20(𝐼) such that 𝑢,𝜙=(𝛼𝑣,𝜙),𝜙𝐻2𝑣(0,1),(2.7)=+𝑏𝑢,𝜙(𝑓,𝜙),𝜙𝐻2(0,1).(2.8)

2.2. The Petrov-Galerkin Formulation

The Petrov-Galerkin formulation corresponding to the above weak formulation (2.7) and (2.8) is defined, respectively, as follows.

Find {𝑢,𝑣}0𝑆,3 such that 𝑢,𝜙=𝛼𝑣,𝜙,𝜙𝑆,1,𝑣+𝑏𝑢,𝜙=𝑓,𝜙,𝜙𝑆,1.(2.9) The integrals in the above Petrov-Galerkin formulation are not evaluated exactly at the implementation level. We, therefore, define the following discrete Petrov-Galerkin procedure in which the integrals are replaced by the Gaussian quadrature in the scheme as follows.

2.3. Discrete Petrov-Galerkin Formulation

The discrete Petrov-Galerkin formulation corresponding to (2.7) and (2.8) is defined, respectively, as follows.

Find {𝑢,𝑣}0𝑆,3 such that 𝑢,𝜙=𝛼𝑣,𝜙,𝜙𝑆,1,𝑣(2.10)+𝑏𝑢,𝜙=𝑓,𝜙,𝜙𝑆,1.(2.11) The approximate solutions 𝑢 and 𝑣 without any conditions on boundary points are expressed as a linear combination of the B-splines as follows: 𝑢(𝑥)=𝑛+1𝑗=1𝛾𝑗𝐵𝑗(𝑥),𝑣(𝑥)=𝑛+1𝑗=1𝛿𝑗𝐵𝑗(𝑥),(2.12) where the 𝑗th basis 𝐵𝑗(𝑥) of the cubic B-splines space 𝑆,3 for 𝑗=1,0,1,2,,𝑛,𝑛+1 is given below: 𝐵𝑗(𝑥)=0,if𝑥𝑥𝑗2,163𝑥𝑥𝑗23,if𝑥𝑗2𝑥𝑥𝑗1,1633+32𝑥𝑥𝑗1+3𝑥𝑥𝑗123𝑥𝑥𝑗13,if𝑥𝑗1𝑥𝑥𝑗,1633+32𝑥𝑗+1𝑥𝑥+3𝑗+1𝑥2𝑥3𝑗+1𝑥3,if𝑥𝑗𝑥𝑥𝑗+1,163𝑥𝑗+2𝑥3,if𝑥𝑗+1𝑥𝑥𝑗+2,0,if𝑥𝑥𝑗+2.(2.13)

For 𝑗=1,0 and 𝑗=𝑛,𝑛+1, the basis functions are defined as in the above form, after extending the partition by introducing fictitious nodal points 𝑥3,𝑥2,𝑥1 on the left-hand side and 𝑥𝑛+1,𝑥𝑛+2,𝑥𝑛+3 on the right-hand side, respectively. Further, the 𝑖th basis 𝜙𝑖(𝑥) of the piecewise linear “hat” splines space 𝑆,1 for 𝑖=0,1,2,,𝑛 is given below: 𝜙𝑖(𝑥)=0,if𝑥𝑥𝑖1,1𝑥𝑥𝑖1,if𝑥𝑖1𝑥𝑥𝑖,1𝑥𝑖+1𝑥,if𝑥𝑖𝑥𝑥𝑖+1,0,if𝑥𝑥𝑖+1.(2.14)

In a similar manner, for 𝑖=0 and 𝑖=𝑛, the basis functions are defined as in the above form, after extending the partition by introducing fictitious nodal point 𝑥1 on the left-hand side and 𝑥𝑛+1 on the right-hand side, respectively. The mixed discrete Petrov-Galerkin method for (2.10) and (2.11) without assuming boundary conditions in the trial space is given as follows: 𝑛+1𝑗=1𝛾𝑗𝐵𝑗,𝜙𝑖𝑛+1𝑗=1𝛿𝑗𝛼𝐵𝑗,𝜙𝑖=0,𝑖=0,1,2,,𝑛,𝑛+1𝑗=1𝛾𝑗𝑏𝐵𝑗,𝜙𝑖+𝑛+1𝑗=1𝛿𝑗𝐵𝑗,𝜙𝑖=𝑓,𝜙𝑖,𝑖=0,1,2,,𝑛,(2.15) with the corresponding equations: 𝑛+1𝑗=1𝛾𝑗𝐵𝑗(0)=0,𝑛+1𝑗=1𝛾𝑗𝐵𝑗(1)=0,𝑛+1𝑗=1𝛿𝑗𝐵𝑗(0)=0,𝑛+1𝑗=1𝛿𝑗𝐵𝑗(0)=0,(2.16) referring to the zero-boundary conditions: 𝑢(0)=0,𝑢(1)=0,𝑣(0)=0,𝑣(1)=0.(2.17) The above set of equations (2.15)–(2.16) can be written as a set of 2𝑛+6 equations in 2𝑛+6 unknowns. Here, we study the effect of quadrature rule in the error analysis. Since we compute the approximations for the solution 𝑢(𝑥) as well as for its second derivative 𝑣(𝑥) with integrals replaced by Gaussian quadrature rule in the formulation, this work may be considered as a quadrature-based mixed Petrov-Galerkin method.

3. Overview of Discrete Petrov-Galerkin Method

Here, the integrals are replaced by composite two-point Gauss rule. Therefore, the resulting method may be described as a “qualocation” approximation, that is, a quadrature-based modification of the collocation method. Further, it may be considered as a Petrov-Galerkin method with a quadrature rule because the test space and trial space are different. Hence, it may be referred to as discrete Petrov-Galerkin method. One practical advantage of this procedure over the orthogonal spline collocation method described in Douglas Jr. and Dupont [1, 2] is that for a given partition there are only half the number of unknowns, and therefore it reduces the size of the matrix.

The qualocation method was first introduced and analysed by Sloan [3] for boundary integral equation on smooth curves. Later on Sloan et al. [4] extended this method to a class of linear second-order two-point boundary value problems and derived optimal error estimates without quasi-uniformity assumption on the finite element mesh. Then, Jones Doss and Pani [5] discussed the qualocation method for a second-order semilinear two-point boundary value problem. Further, Pani [6] expanded its scope by adapting the analysis to a semilinear parabolic initial and boundary value problem in a single space variable. Jones Doss and Pani [7] extended this method to the free boundary problem, that is, one-dimensional single-phase Stefan problem for which part of the boundary has to be found out along with the solution process. A quadrature-based Petrov-Galerkin method applied to higher dimensional boundary value problems is studied in Bialecki et al. [8, 9] and Ganesh and Mustapha [10].

The main idea of this paper is that a quadrature based approximation for a fourth order problem is analyzed in mixed Galerkin setting. The organization of this paper is as follows. In previous Sections 1 and 2, the problem is introduced; the weak and the Galerkin formulations are defined. Overview of discrete Petrov-Galerkin method is discussed in Section 3. Preliminaries required for our analysis are mentioned in Section 4. Error analysis is carried over in Section 5. Throughout this paper 𝐶 is a generic positive constant, whose dependence on the smoothness of the exact solution can be easily determined from the proofs.

4. Preliminaries

We assume that 𝛼 and 𝑏 are such that 𝛼,𝑏𝐶4𝐼,(4.1) where 𝐼=[0,1]. We assume that the problem consisting of the coupled equations (1.4) and (1.5) is uniquely solvable for a given sufficiently smooth function 𝑓(𝑥). It can be proved that the quadrature rule in (2.3) has an error bound of the form 𝐸||||𝒬(𝑔)=𝑔||||(𝑔)𝐶𝑛𝑖=14𝑘𝑔(4)𝐿1(𝐼𝑘).(4.2) This follows from Peano’s kernel theorem (see [11]).

The following inequality is frequently used in our analysis. If 𝑣𝑊𝑚𝑝(𝐸) with 𝑝[1,], then there exists a positive constant 𝐶 depending only on 𝑚 such that, for any 𝛿satisfying 0<𝛿|𝐸|1,𝑣𝑊𝑖𝑝(𝐸)𝛿𝐶𝑚𝑖𝑣𝑊𝑚𝑝(𝐸)+𝛿𝑖𝑣𝐿𝑝(𝐸),0𝑖𝑚1,(4.3) where |𝐸|denotes the length of 𝐸. For a detailed proof, one may refer to appendix of Sloan et al. [4] or Chapter 4 of Adams [12]. Let us use the following notation: 𝐿𝑣=𝑣.(4.4) The adjoint operator 𝐿 with corresponding adjoint boundary condition is defined as follows: 𝐿𝜙=𝜙,𝜙(0)=𝜙(1)=0.(4.5) Since 𝐿 is a self-adjoint operator, we mention below the regularity of 𝐿(equal to 𝐿) in the 𝑞 norm. We make a stronger assumption as in Sloan et al. [4] that for arbitrary 𝑞[1,], there exists a positive constant 𝐶 such that 𝐿𝑢𝐿𝑞(𝐼)𝐶𝑢𝑊2𝑞(𝐼).(4.6) We have the following inequality due to the Sobolev embedding theorem; the proof of which can be found in page 97, Adams [12], 𝜙𝐿(𝐼𝑘)𝜙𝑊1𝑝(𝐼𝑘);1𝑝,𝜙𝑊1𝑝𝐼𝑘.(4.7)

5. Convergence Analysis

Hereafter throughout this section, for 𝑝 and 𝑞 with 1𝑝,𝑞,s  and 𝑝1+𝑞1=1, we use the following notations: 𝑣0,𝑝=𝑣𝐿𝑝,𝑣𝑠,𝑝=𝑣𝑊𝑠𝑝,𝑣𝑠,𝑝,𝑘=𝑣𝑊𝑠𝑝(𝐼𝑘).(5.1) Let us denote the error between 𝑢 and 𝑢 by 𝜀 and the error between 𝑣 and 𝑣 by 𝑒, respectively, that is, 𝜀=𝑢𝑢 and 𝑒=𝑣𝑣. Using (2.11) and (1.5), we obtain the following error equations: 𝑒,𝜙=𝑣𝑣,𝜙=𝑣,𝜙𝑓𝑏𝑢,𝜙𝑏=𝑢𝑢,𝜙=𝑏𝜀,𝜙,(5.2) and therefore we get 𝑒,𝜙=𝑏𝜀,𝜙,𝜙𝑆,1.(5.3) Further, using (2.10) and (1.4),𝜀,𝜙=𝑢𝑢,𝜙=𝛼𝑣𝑣,𝜙=𝛼𝑒,𝜙,(5.4) and therefore we have 𝜀,𝜙=𝛼𝑒,𝜙,𝜙𝑆,1.(5.5) The following lemma gives estimates for the error in the quadrature rule for the term (𝑒𝜒) and (𝜀𝜒) for 𝜒𝑆,1. These estimates are required for our error analysis later. The proof of the lemma is similar to the proof of Lemma 4.2 of Sloan et al. [4].

Lemma 5.1. For all 𝜒𝑆,1 and h sufficiently small, (a)𝐸(𝑒𝜒)𝐶4𝑣6,𝑝𝜒1,𝑞, (b)𝐸(𝑒𝜒)𝐶3𝑣6,𝑝𝜒0,𝑞, (c)𝐸(𝜀𝜒)𝐶4𝑢6,𝑝𝜒1,𝑞, (d)𝐸(𝜀𝜒)𝐶3𝑢6,𝑝𝜒0,𝑞.

The following result gives estimate for 𝜀(𝑥), where 𝑥 is any arbitrary point in 𝐼. This estimate is crucial for our error analysis.

Lemma 5.2. Let 𝑢 be the weak solution of (1.4) defined through (2.7). Further, let 𝑢 be the corresponding discrete Petrov-Galerkin solution defined through (2.10). Then, the error 𝜀=𝑢𝑢 satisfies ||𝜀𝑥||𝐶2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝,(5.6) where 𝑥 is an arbitrary point in [0,1].

Proof. For a given 𝑥[0,1], let Φ be an element of 𝐿𝑝(𝐼)𝐶(𝐼) satisfying the following auxiliary problem: Φ=0,𝑥𝐼𝑥,Φ(0)=Φ(1)=0,Φ𝑥Φ+𝑥=1.(5.7) The above problem has a solution. For example, Φ(𝑥)=𝑥1𝑥,0𝑥𝑥,𝑥(𝑥1),𝑥𝑥1(5.8) satisfies the above differential equation, the boundary conditions, and the jump condition.
Let us define Ψ as follows: ΦΨ(𝑥)=,𝑥𝐼𝑥,0,at𝑥=𝑥.(5.9) Then, Ψ=0  a.e. on 𝐼. We first multiply 𝜀 with Ψ and then integrate over 𝐼. On applying integration by parts, using the fact that 𝜀(0)=𝜀(1)=0 and the jump condition for Φ, we obtain 𝜀0==,Ψ𝑥0𝜀Ψ+1𝑥𝜀Ψ=𝑥0𝜀Φ+1𝑥𝜀Φ=𝜀Φ𝑥0𝑥0𝜀Φ+𝜀Φ1𝑥1𝑥𝜀Φ=𝜀𝑥Φ𝑥Φ+𝑥𝑥0𝜀Φ1𝑥𝜀Φ=𝜀𝑥𝑥0𝜀Φ1𝑥𝜀Φ.(5.10) Applying integration by parts once again, using boundary condition for Φ and the continuity of Φ, we obtain 0=𝜀𝑥𝜀Φ𝑥0𝑥0𝜀𝜀Φ+Φ1𝑥1𝑥𝜀Φ=𝜀𝑥+𝜀,Φ,(5.11) that is, 𝜀(𝑥)=(𝜀,Φ). Let Φ be the linear interpolant of Φ. Then, we have 𝜀𝑥=𝜀,ΦΦ+𝜀,Φ𝜀,Φ+𝜀,Φ||𝜀𝑥||||𝜀,ΦΦ||+||𝐸𝜀Φ||+||𝜀,Φ||𝑇1+𝑇2+𝑇3.(5.12) We know that Φ1,𝑞ΦΦ1,𝑞+Φ1,𝑞𝐶Φ2,𝑞+Φ2,𝑞𝐶Φ2,𝑞.(5.13) We now compute the estimates for the terms 𝑇1, 𝑇2, and 𝑇3 as follows: 𝑇1=||𝜀,ΦΦ||𝜀0,𝑝ΦΦ0,𝑞𝐶2𝜀2,𝑝Φ2,𝑞.(5.14) Using Lemma 5.1(c) and (5.13), we obtain 𝑇2=||𝐸𝜀Φ||𝐶4𝑢6,𝑝Φ2,𝑞.(5.15) Using (5.5), (2.3), and the Sobolev embedding theorem (4.7) locally on 𝐼𝑘 for both 𝑒0,,𝑘 and Φ0,,𝑘, we have 𝑇3=||𝜀,Φ||=||𝛼𝑒,Φ||𝐶𝑛𝑘=1𝑘2𝑒0,,𝑘Φ0,,𝑘𝐶𝑛𝑘=1𝑘2𝑒1,𝑝,𝑘Φ1,𝑞,𝑘.(5.16) Using Hölder's inequality for sums and (5.13), we have 𝑇3𝑒𝐶1,𝑝Φ1,𝑞𝑒𝐶1,𝑝Φ2,𝑞.(5.17) For Φ satisfying the auxiliary problem, it is easy to verify that Φ2,𝑞𝐾, where 𝐾 is a constant not depending on .
Using 𝑇1, 𝑇2, and 𝑇3 in (5.12), we have ||𝜀𝑥||𝐶2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝.(5.18) This completes the proof.

In the following lemma, we initially compute the error (𝑣𝑣) in terms of (𝑢𝑢), and then later on we establish an optimal estimate of error (𝑣𝑣) independent of (𝑢𝑢).

Lemma 5.3. Let 𝑢 and 𝑣 be the weak solutions of the coupled equations (1.4) and (1.5) defined through (2.7) and (2.8), respectively. Further, let 𝑢 and 𝑣 be the corresponding discrete Petrov-Galerkin solutions defined through (2.10) and (2.11), respectively. Then the estimates of the errors 𝑒=𝑣𝑣 in 𝐿𝑝, 𝑊1𝑝, and 𝑊2𝑝 norms are given as follows: 𝑒0,𝑝𝐶4𝑣6,𝑝+5𝑢6,𝑝+3𝜀2,𝑝,𝑒1,𝑝𝐶3𝑣6,𝑝+4𝑢6,𝑝+2𝜀2,𝑝,𝑒2,𝑝𝐶2𝑣6,𝑝+4𝑢6,𝑝+2𝜀2,𝑝.(5.19)

Proof. Let 𝜂 be an arbitrary element of 𝐿𝑞, and let 𝜙𝑊2𝑞 be the solution of the auxiliary problem 𝐿𝜙=𝜂,𝜙(0)=𝜙(1)=0.(5.20) We now have 𝑒=𝑒,𝜂,𝐿𝜙=𝐿𝑒=𝑒,𝜙,𝜙𝜙+𝑒,𝜙=𝑒,𝜙𝜙+𝑒,𝜙𝑒,𝜙+𝑒,𝜙=𝑒,𝜙𝜙+𝐸𝑒𝜙+𝑒,𝜙,||𝑒||||𝑒,𝜂,𝜙𝜙||+||𝐸𝑒𝜙||+||𝑒,𝜙||𝑇4+𝑇5+𝑇6,(5.21) where 𝜙𝑆,1 is the linear interpolant of 𝜙.
We know that 𝜙1,𝑞𝜙𝜙1,𝑞+𝜙1,𝑞𝐶𝜙2,𝑞+𝜙2,𝑞𝐶𝜙2,𝑞.(5.22) We shall compute the estimates for the terms 𝑇4, 𝑇5, and 𝑇6 as follows: 𝑇4=||𝑒,𝜙𝜙||𝑒0,𝑝𝜙𝜙0,𝑞𝐶2𝑒2,𝑝𝜙2,𝑞,𝑇5=||𝐸𝑒𝜙||𝐶4𝑣6,𝑝𝜙1,𝑞𝐶4𝑣6,𝑝𝜙2,𝑞byLemma5.1(a),(5.22).(5.23) Using (5.3), (2.3), and the Sobolev embedding theorem (4.7) locally on 𝐼𝑘 for 𝜙0,,𝑘, we have 𝑇6=||𝑒,𝜙||=||𝑏𝜀,𝜙||𝐶𝑛𝑘=1𝑘2𝜀0,,𝑘𝜙0,,𝑘𝐶𝑛𝑘=1𝑘2𝜀0,,𝑘𝜙1,𝑞,𝑘.(5.24) Using Hölder's inequality for sums, Lemma 5.2, and (5.22), we obtain 𝑇6𝐶2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝𝜙1,𝑞𝐶3𝜀2,𝑝+5𝑢6,𝑝+2𝑒1,𝑝𝜙2,𝑞.(5.25) Substituting 𝑇4, 𝑇5, and 𝑇6 in (5.21), we have ||𝑒||,𝜂𝐶2𝑒2,𝑝+4𝑣6,𝑝+3𝜀2,𝑝+5𝑢6,𝑝+2𝑒1,𝑝𝜙2,𝑞.(5.26) Using (4.6) and the regularity of the auxiliary problem, we have 𝜙2,𝑞𝐶𝜂0,𝑞. Since 𝜂𝐿𝑞 is arbitrary, we have 𝑒0,𝑝𝐶2𝑒2,𝑝+3𝜀2,𝑝+4𝑣6,𝑝+5𝑢6,𝑝.(5.27) We now estimate 𝑒 via a projection argument. Let 𝑃 be the orthogonal projection onto 𝑆,1 with respect to 𝐿2 inner product defined by 𝑣𝑃𝑣,𝜓=0,𝜓𝑆,1.(5.28) The domain of 𝑃 may be taken to be 𝐿1. From Crouzeix and Thomée [13] and de Boor [14], it is seen that the 𝐿2 projection is stable. Thus, 𝑃𝑣0,𝑝𝐶𝑣0,𝑝.(5.29) Then the error 𝑒 can be interpreted in terms of the error of the above projection: 𝑒0,𝑝=𝑣𝑣0,𝑝𝑣𝑃𝑣0,𝑝+𝑃𝑣𝑣0,𝑝.(5.30) From the stability property (5.29), the error in the projection follows as in de Boor [14], that is, 𝑣𝑃𝑣0,𝑝𝐶2𝑣2,𝑝𝐶2𝑣4,𝑝.(5.31) Then the remaining task is to compute the estimate of 𝑃𝑣𝑣0,𝑝.
For 𝜓𝑆,1, 𝑃𝑣𝑣,𝜓=𝑃𝑣𝑣+𝑣𝑣,𝜓=𝑃𝑣𝑣,𝜓+𝑣𝑣,𝜓=𝑣𝑣,𝜓𝑃using(5.28),𝑣𝑣,𝜓=𝑒,𝜓=𝑒,𝜓𝑒,𝜓+𝑒,𝜓=𝐸𝑒𝜓+𝑒,𝜓,||𝑃𝑣𝑣,𝜓||||𝐸𝑒𝜓||+||𝑒,𝜓||𝑇7+𝑇8.(5.32) We shall compute the estimates for the terms 𝑇7 and 𝑇8𝑇7=||𝐸𝑒𝜓||𝐶3𝑣6,𝑝𝜓0,𝑞(5.33) by Lemma 5.1(b).
Following the steps of computation involved in the term 𝑇6, we obtain the estimate of 𝑇8 as 𝑇8=||𝑒,𝜓||𝐶2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝𝜓0,𝑞,(5.34) where we have used the inverse inequality 𝜓1,𝑞,𝑘𝑘1𝜓0,𝑞,𝑘 locally. Using 𝑇7 and 𝑇8 in (5.32), we get ||𝑃𝑣𝑣,𝜓||𝐶3𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝𝜓0,𝑞.(5.35) We now show the above inequality for 𝜂𝐿𝑞 to obtain 𝑃𝑣𝑣0,𝑝.
Now let 𝜂 be an arbitrary element of 𝐿𝑞. Then since 𝑣𝑆,1, it follows from the definition of 𝑃𝜂, (5.35), and (5.29) with 𝑝 replaced by 𝑞, that 𝑃0=𝑣𝑣,𝜂𝑃𝜂,||𝑃𝑣𝑣||=||𝑃,𝜂𝑣𝑣,𝑃𝜂||𝐶3𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝𝑃𝜂0,𝑞𝐶3𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝𝜂0,𝑞,𝑃𝑣𝑣0,𝑝𝐶3𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝.(5.36) Now, from (5.30), (5.31), and (5.36), we conclude that 𝑒0,𝑝𝐶2𝑣4,𝑝+𝐶3𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝𝐶2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝.(5.37) Now, using the fact 𝑒2,𝑝𝑒1,𝑝+𝑒0,𝑝 and the above estimate, we have 𝑒2,𝑝𝑒1,𝑝+𝐶2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝𝑒𝐶1,𝑝+2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝑒𝐶1,𝑝+2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝.(5.38) Now using (4.3) with 𝑚=2 and 𝑖=1, we have 𝑒1,𝑝𝐶1𝑒0,𝑝𝑒+2,𝑝.(5.39) Substituting (5.39) in the above expression, we obtain 𝑒2,𝑝𝐶1𝑒0,𝑝𝑒+2,𝑝+2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝.(5.40) For sufficiently small , we have 𝑒2,𝑝𝐶1𝑒0,𝑝+2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝.(5.41) Using (5.41) in (5.27), 𝑒0,𝑝𝐶21𝑒0,𝑝+2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝+4𝑣6,𝑝+5𝑢6,𝑝+3𝜀2,𝑝.(5.42) For sufficiently small , we get 𝑒0,𝑝𝐶4𝑣6,𝑝+5𝑢6,𝑝+3𝜀2,𝑝.(5.43) Using (5.43) in (5.41), we have 𝑒2,𝑝𝐶14𝑣6,𝑝+5𝑢6,𝑝+3𝜀2,𝑝+2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝐶2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝.(5.44) Using (5.43) and (5.44) in (5.39), we have 𝑒1,𝑝𝐶14𝑣6,𝑝+5𝑢6,𝑝+3𝜀2,𝑝+2𝑣6,𝑝+2𝜀2,𝑝+4𝑢6,𝑝𝐶3𝑣6,𝑝+4𝑢6,𝑝+2𝜀2,𝑝.(5.45) Equations (5.43), (5.44), and (5.45) give the required result.

We now compute the error estimate of 𝜀 in 𝐿𝑝,𝑊1𝑝,and 𝑊2𝑝 norms as has been done in the previous case.

Lemma 5.4. Let 𝑢 and 𝑣 be the weak solutions of the coupled equations (1.4) and (1.5) defined through (2.7) and (2.8), respectively. Further, let 𝑢 and 𝑣 be the corresponding discrete Petrov-Galerkin solutions defined through (2.10) and (2.11), respectively. Then the estimates of the errors 𝜀=𝑢𝑢 in 𝐿𝑝,𝑊1𝑝and 𝑊2𝑝 norms are given as follows: 𝜀0,𝑝𝐶4𝑢6,𝑝𝑒+1,𝑝,𝜀1,𝑝𝐶3𝑢6,𝑝+𝑒1,𝑝,𝜀2,𝑝𝐶2𝑢6,𝑝+𝑒1,𝑝.(5.46)

Proof. Let 𝜌 be an arbitrary element of 𝐿𝑞, and let 𝜙𝑊2𝑞 be the unique solution of the auxiliary problem 𝐿𝜙=𝜌,𝜙(0)=𝜙(1)=0.(5.47) Then we have 𝜀=𝜀,𝜌,𝐿𝜙=𝐿𝜀=𝜀,𝜙=𝜀,𝜙,𝜙𝜙+𝜀,𝜙𝜀,𝜙+𝜀,𝜙,(5.48) where 𝜙𝑆,1 is a linear interpolant of 𝜙, ||𝜀,𝜌||||𝜀,𝜙𝜙||+||𝐸𝜀𝜙||+||𝜀,𝜙||𝑇9+𝑇10+𝑇11.(5.49) Following the steps involved in the computation of 𝑇4 and 𝑇5, we obtain the estimates of 𝑇9 and 𝑇10 as follows: 𝑇9𝐶2𝜀2,𝑝𝜙2,𝑞,𝑇10𝐶4𝑢6,𝑝𝜙2,𝑞,(5.50) by Lemma 5.1(c) and (5.22).
Using (5.5) and (2.3) first, then the Sobolev embedding theorem (4.7) locally on 𝐼𝑘 for 𝜙0,,𝑘 and 𝑒0,,𝑘 to estimate 𝑇11, we have 𝑇11=||𝜀,𝜙||=||𝛼𝑒,𝜙||𝐶𝑛𝑘=1𝑘2𝑒0,,𝑘𝜙0,,𝑘𝐶𝑛𝑘=1𝑘2𝑒0,,𝑘𝜙1,𝑞,𝑘𝐶𝑛𝑘=1𝑘2𝑒1,𝑝,𝑘𝜙1,𝑞,𝑘.(5.51) Further, using Hölder's inequality for sums and (5.22), we obtain 𝑇11𝑒𝐶1,𝑝𝜙1,𝑞𝑒𝐶1,𝑝𝜙2,𝑞.(5.52) Substituting the estimates 𝑇9, 𝑇10, and 𝑇11 in (5.49), we obtain ||𝜀,𝜌||𝐶2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝𝜙2,𝑞.(5.53) Using (4.6) and regularity of the auxiliary problem, we have 𝜙2,𝑞𝐶𝜌𝑜,𝑞. Since 𝜌𝐿𝑞 is arbitrary, we have 𝜀0,𝑝𝐶2𝜀2,𝑝+4𝑢6,𝑝𝑒+1,𝑝.(5.54) The estimate of 𝜀0,𝑝 can be obtained through a projection argument as mentioned in Lemma 5.3 as 𝜀0,𝑝𝐶2𝑢6,𝑝+𝑒1,𝑝,(5.55) where we have used Lemma 5.1(d). In a similar manner we can compute the estimates for 𝜀0,𝑝, 𝜀1,𝑝 and 𝜀2,𝑝 as 𝜀0,𝑝𝐶4𝑢6,𝑝𝑒+1,𝑝,𝜀1,𝑝𝐶3𝑢6,𝑝+𝑒1,𝑝,𝜀2,𝑝𝐶2𝑢6,𝑝+𝑒1,𝑝.(5.56) Using all the estimates from Lemmas 5.3 and 5.4, we have the following main error estimates.

Theorem 5.5. Assume that 𝑢 and 𝑣 satisfy (1.4) and (1.5), respectively, with (4.1). Assume also that 𝑢𝑊6𝑝 and 𝑣𝑊6𝑝, where 𝑝[1,]. Then (2.10) and (2.11) have unique solutions 𝑢0𝑆,3 and 𝑣0𝑆,3, respectively, and for sufficiently small, one has 𝑢𝑢𝑖,𝑝𝐶4𝑖𝑢6,𝑝+𝑣6,𝑝,𝑣𝑣𝑖,𝑝𝐶4𝑖𝑢6,𝑝+𝑣6,𝑝,𝑖=0,1,2.(5.57)

Proof. Assume temporarily that solutions 𝑢 and 𝑣 of (2.10) and (2.11), respectively, exist. Using (5.46) in (5.45), we obtain 𝑒1,𝑝𝐶3𝑣6,𝑝+4𝑢6,𝑝+22𝑢6,𝑝+𝑒1,𝑝.(5.58) For sufficiently small , we have 𝑒1,𝑝𝐶3𝑣6,𝑝+4𝑢6,𝑝.(5.59) An application of the above in (5.46), we get 𝜀2,𝑝𝐶2𝑢6,𝑝+3𝑣6,𝑝.(5.60) Apply (5.59) in (5.56) to have 𝜀0,𝑝𝐶4𝑢6,𝑝+4𝑣6,𝑝.(5.61) Use (5.60) in (5.43) to get 𝑒0,𝑝𝐶4𝑣6,𝑝+5𝑢6,𝑝.(5.62) Using (5.60) in (5.44), we obtain 𝑒2,𝑝𝐶2𝑣6,𝑝+4𝑢6,𝑝.(5.63) Using (5.61) and (5.60) in (5.39) with 𝑒 replaced by 𝜀, we have 𝜀1,𝑝𝐶3𝑢6,𝑝+3𝑣6,𝑝.(5.64) The required result can be obtained from estimates (5.59) to (5.64).

So far we have assumed temporarily that solutions 𝑢 and 𝑣 exist. We now discuss the existence and uniqueness of discrete Petrov-Galerkin approximation. Since the matrix corresponding to (2.10) and (2.11) with zero boundary conditions for 𝑢 and 𝑣 is square, existence of 𝑢0𝑆,3 and 𝑣0𝑆,3 for any 𝑓𝐶0(𝐼) will follow from uniqueness, that is, from the property that the corresponding homogeneous equations have only trivial solutions.

Suppose that 𝑢 and 𝑣 corresponding to 𝑢 and 𝑣 satisfy 𝑢𝛼𝑣,𝜒𝑣=0,+𝑏𝑢,𝜒=0,𝜒𝑆,1.(5.65) It follows from (5.61) and (5.62) (with 𝑢 replaced by 0 and eventually 𝑣0) that, for sufficiently small , 𝑢0,𝑝𝑣0,0,𝑝0,(5.66) and hence 𝑢0 and 𝑣0. Thus, uniqueness is proved, and hence existence follows from uniqueness.

References

  1. J. Douglas, Jr. and T. Dupont, “A finite element collocation method for quasilinear parabolic equations,” Mathematics of Computation, vol. 27, pp. 17–28, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. J. Douglas, Jr. and T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable, vol. 385 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1974.
  3. I. H. Sloan, “A quadrature-based approach to improving the collocation method,” Numerische Mathematik, vol. 54, no. 1, pp. 41–56, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. I. H. Sloan, D. Tran, and G. Fairweather, “A fourth-order cubic spline method for linear second-order two-point boundary value problems,” IMA Journal of Numerical Analysis, vol. 13, no. 4, pp. 591–607, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. L. Jones Doss and A. K. Pani, “A qualocation method for a semilinear second-order two-point boundary value problem,” in Functional Analysis with Current Applications in Science, Technology and Industry, M. Brokate and A. H. Siddiqi, Eds., vol. 377 of Pitman Research Notes in Mathematics, pp. 128–144, Addison Wesley Longman, Harlow, UK, 1998. View at: Google Scholar | Zentralblatt MATH
  6. A. K. Pani, “A qualocation method for parabolic partial differential equations,” IMA Journal of Numerical Analysis, vol. 19, no. 3, pp. 473–495, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. L. Jones Doss and A. K. Pani, “A qualocation method for a unidimensional single phase semilinear Stefan problem,” IMA Journal of Numerical Analysis, vol. 25, no. 1, pp. 139–159, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. B. Bialecki, M. Ganesh, and K. Mustapha, “A Petrov-Galerkin method with quadrature for elliptic boundary value problems,” IMA Journal of Numerical Analysis, vol. 24, no. 1, pp. 157–177, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. B. Bialecki, M. Ganesh, and K. Mustapha, “An ADI Petrov-Galerkin method with quadrature for parabolic problems,” Numerical Methods for Partial Differential Equations, vol. 25, no. 5, pp. 1129–1148, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. M. Ganesh and K. Mustapha, “A Crank-Nicolson and ADI Galerkin method with quadrature for hyperbolic problems,” Numerical Methods for Partial Differential Equations, vol. 21, no. 1, pp. 57–79, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, NY, USA, 2nd edition, 1975.
  12. R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1975. View at: Zentralblatt MATH
  13. M. Crouzeix and V. Thomée, “The stability in Lp and Wp1 of the L2-projection onto finite element function spaces,” Mathematics of Computation, vol. 48, no. 178, pp. 521–532, 1987. View at: Publisher Site | Google Scholar | MathSciNet
  14. C. de Boor, “A bound on the L-norm of L2-approximation by splines in terms of a global mesh ratio,” Mathematics of Computation, vol. 30, no. 136, pp. 765–771, 1976. View at: Google Scholar

Copyright © 2012 L. Jones Tarcius Doss and A. P. Nandini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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