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 | 18 pages |

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

Academic Editor: Attila GilΓ‘nyi
Received20 Jul 2011
Revised24 Nov 2011
Accepted25 Nov 2011
Published13 Feb 2012


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: 𝑑2𝑑π‘₯2ξ‚Έπ‘‘π‘Ž(π‘₯)2𝑒𝑑π‘₯2ξ‚Ή+𝑏(π‘₯)𝑒=𝑓(π‘₯),π‘₯∈𝐼=(0,1),(1.1) subject to the boundary conditions 𝑒(0)=0,𝑒(1)=0;π‘’ξ…žξ…ž(0)=0,π‘’ξ…žξ…ž(1)=0,(1.2) where π‘Ž(π‘₯)β‰ 0,π‘₯∈𝐼. Let 𝛼(π‘₯)=1/π‘Ž(π‘₯). We, hereafter, suppress the dependency of the independent variable π‘₯ on the functions 𝛼(π‘₯), 𝑏(π‘₯), and 𝑓(π‘₯). 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 π‘’ξ…žξ…ž=𝛼𝑣,π‘₯∈𝐼.(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: π‘’ξ…žξ…žπ‘£=𝛼𝑣,π‘₯∈𝐼,with𝑒(0)=𝑒(1)=0,(1.4)ξ…žξ…ž+𝑏𝑒=𝑓,π‘₯∈𝐼,with𝑣(0)=𝑣(1)=0.(1.5) In this paper, the error analysis will take place in the usual Sobolev space π‘Šπ‘šπ‘(𝐼) defined on the domain 𝐼=(0,1) with π»π‘š(𝐼) denoting π‘Šπ‘š2(𝐼). The Sobolev norms are given below. For an open interval 𝐸 and a non negative integer π‘š, β€–π‘£β€–π‘Šπ‘šπ‘(𝐸)=ξƒ©π‘šξ“π‘–=0‖‖𝑣(𝑖)‖‖𝑝𝐿𝑝(𝐸)ξƒͺ1/𝑝,if1≀𝑝<∞,=max1≀𝑖≀𝑛‖‖𝑣(𝑖)β€–β€–πΏβˆž(𝐸),if𝑝=∞.(1.6) We suppress the dependence of the norms on 𝐼 when 𝐸=𝐼. Further, π»π‘š0(𝐼) denotes the function space {πœ™βˆˆπ»π‘š(𝐼)βˆΆπœ™(0)=πœ™(1)=0}.(1.7)

2. Continuous and Discrete 𝐻1-Galerkin Formulation

Given 𝑛>1, let Ξ π‘›βˆΆ0=π‘₯0<π‘₯1<β‹―<π‘₯𝑛=1(2.1) be an arbitrary partition of [0,1] with the property that β„Žβ†’0 as π‘›β†’βˆž, where β„Ž=max1β‰€π‘˜β‰€π‘›β„Žπ‘˜ and β„Žπ‘˜=π‘₯π‘˜βˆ’π‘₯π‘˜βˆ’1,π‘˜=1,…,𝑛. Let (𝑒,𝑣) represent the 𝐿2 inner product, and let βŸ¨π‘’,π‘£βŸ©β„Ž represent the discrete inner product of any two functions 𝑒,π‘£βˆˆπΏ2(𝐼) and be defined as follows: (ξ€œπ‘’,𝑣)=𝑒𝑣𝑑π‘₯,βŸ¨π‘’,π‘£βŸ©β„Ž=π’¬β„Ž(𝑒𝑣),(2.2) where π’¬β„Ž is the fourth-order Gaussian quadrature rule: π’¬β„Ž1(𝑔)∢=2𝑛𝑖=1β„Žπ‘˜ξ€Ίπ‘”ξ€·π‘₯π‘˜,1ξ€Έξ€·π‘₯+π‘”π‘˜,2ξ€Έξ€».(2.3) Here, π‘₯π‘˜,𝑖=π‘₯π‘˜βˆ’1+πœ‰π‘–β„Žπ‘˜,𝑖=1,2, are the two Gaussian points in the subinterval [π‘₯π‘˜βˆ’1,π‘₯π‘˜] with πœ‰1√=(1/2)(1βˆ’1/3), πœ‰2=1βˆ’πœ‰1.

Let us now consider the following cubic spline space as trial space: π‘†β„Ž,3=ξ€½πœ‘βˆˆπΆ2(𝐼)βˆΆπœ‘|πΌπ‘˜βˆˆπ‘ƒ3ξ€·πΌπ‘˜ξ€Έξ€Ύ,π‘˜=1,2,…,𝑛,(2.4) where π‘ƒπ‘Ÿ(πΌπ‘˜) is the space of polynomials of degree π‘Ÿ defined over the π‘˜th subinterval πΌπ‘˜=[π‘₯π‘˜βˆ’1,π‘₯π‘˜].

The corresponding space with zero Dirichlet boundary condition is denoted by 0π‘†β„Ž,3=ξ€½πœ‘βˆˆπ‘†β„Ž,3ξ€ΎβˆΆπœ‘(0)=πœ‘(1)=0.(2.5) Further, let us consider the following piecewise linear space π‘†β„Ž,1=ξ€½πœ‘βˆˆπΆ(𝐼)βˆΆπœ‘|πΌπ‘˜βˆˆπ‘ƒ1ξ€·πΌπ‘˜ξ€Έξ€Ύ,π‘˜=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 {𝑒,𝑣}∈𝐻20(𝐼) such that ξ€·π‘’ξ…žξ…žξ€Έ,πœ™=(𝛼𝑣,πœ™),πœ™βˆˆπ»2𝑣(0,1),(2.7)ξ…žξ…žξ€Έ=+𝑏𝑒,πœ™(𝑓,πœ™),πœ™βˆˆπ»2(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 {π‘’β„Ž,π‘£β„Ž}∈0π‘†β„Ž,3 such that ξ€·π‘’β„Žξ…žξ…ž,πœ™β„Žξ€Έ=ξ€·π›Όπ‘£β„Ž,πœ™β„Žξ€Έ,πœ™β„Žβˆˆπ‘†β„Ž,1,ξ€·π‘£β„Žξ…žξ…ž+π‘π‘’β„Ž,πœ™β„Žξ€Έ=𝑓,πœ™β„Žξ€Έ,πœ™β„Žβˆˆπ‘†β„Ž,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 {π‘’β„Ž,π‘£β„Ž}∈0π‘†β„Ž,3 such that ξ«π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž=βŸ¨π›Όπ‘£β„Ž,πœ™β„ŽβŸ©β„Ž,πœ™β„Žβˆˆπ‘†β„Ž,1,𝑣(2.10)β„Žξ…žξ…ž+π‘π‘’β„Ž,πœ™β„Žξ¬β„Ž=βŸ¨π‘“,πœ™β„ŽβŸ©β„Ž,πœ™β„Žβˆˆπ‘†β„Ž,1.(2.11) The approximate solutions π‘’β„Ž and π‘£β„Ž without any conditions on boundary points are expressed as a linear combination of the B-splines as follows: π‘’β„Ž(π‘₯)=𝑛+1𝑗=βˆ’1𝛾𝑗𝐡𝑗(π‘₯),π‘£β„Ž(π‘₯)=𝑛+1𝑗=βˆ’1𝛿𝑗𝐡𝑗(π‘₯),(2.12) where the 𝑗th basis 𝐡𝑗(π‘₯) of the cubic B-splines space π‘†β„Ž,3 for 𝑗=βˆ’1,0,1,2,…,𝑛,𝑛+1 is given below: π΅π‘—βŽ§βŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺ⎩(π‘₯)=0,ifπ‘₯≀π‘₯π‘—βˆ’2,16β„Ž3ξ€·π‘₯βˆ’π‘₯π‘—βˆ’2ξ€Έ3,ifπ‘₯π‘—βˆ’2≀π‘₯≀π‘₯π‘—βˆ’1,16β„Ž3ξ‚€β„Ž3+3β„Ž2ξ€·π‘₯βˆ’π‘₯π‘—βˆ’1ξ€Έξ€·+3β„Žπ‘₯βˆ’π‘₯π‘—βˆ’1ξ€Έ2ξ€·βˆ’3π‘₯βˆ’π‘₯π‘—βˆ’1ξ€Έ3,ifπ‘₯π‘—βˆ’1≀π‘₯≀π‘₯𝑗,16β„Ž3ξ‚€β„Ž3+3β„Ž2ξ€·π‘₯𝑗+1ξ€Έξ€·π‘₯βˆ’π‘₯+3β„Žπ‘—+1ξ€Έβˆ’π‘₯2ξ€·π‘₯βˆ’3𝑗+1ξ€Έβˆ’π‘₯3,ifπ‘₯𝑗≀π‘₯≀π‘₯𝑗+1,16β„Ž3ξ€·π‘₯𝑗+2ξ€Έβˆ’π‘₯3,ifπ‘₯𝑗+1≀π‘₯≀π‘₯𝑗+2,0,ifπ‘₯β‰₯π‘₯𝑗+2.(2.13)

For 𝑗=βˆ’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 π‘₯βˆ’3,π‘₯βˆ’2,π‘₯βˆ’1 on the left-hand side and π‘₯𝑛+1,π‘₯𝑛+2,π‘₯𝑛+3 on the right-hand side, respectively. Further, the 𝑖th basis πœ™π‘–(π‘₯) of the piecewise linear β€œhat” splines space π‘†β„Ž,1 for 𝑖=0,1,2,…,𝑛 is given below: πœ™π‘–(⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩π‘₯)=0,ifπ‘₯≀π‘₯π‘–βˆ’1,1β„Žξ€·π‘₯βˆ’π‘₯π‘–βˆ’1ξ€Έ,ifπ‘₯π‘–βˆ’1≀π‘₯≀π‘₯𝑖,1β„Žξ€·π‘₯𝑖+1ξ€Έβˆ’π‘₯,ifπ‘₯𝑖≀π‘₯≀π‘₯𝑖+1,0,ifπ‘₯β‰₯π‘₯𝑖+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 π‘₯βˆ’1 on the left-hand side and π‘₯𝑛+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+1𝑗=βˆ’1π›Ύπ‘—ξ«π΅π‘—ξ…žξ…ž,πœ™π‘–ξ¬β„Žβˆ’π‘›+1𝑗=βˆ’1𝛿𝑗𝛼𝐡𝑗,πœ™π‘–ξ¬β„Ž=0,𝑖=0,1,2,…,𝑛,𝑛+1𝑗=βˆ’1𝛾𝑗𝑏𝐡𝑗,πœ™π‘–ξ¬β„Ž+𝑛+1𝑗=βˆ’1π›Ώπ‘—ξ«π΅π‘—ξ…žξ…ž,πœ™π‘–ξ¬β„Ž=βŸ¨π‘“,πœ™π‘–βŸ©β„Ž,𝑖=0,1,2,…,𝑛,(2.15) with the corresponding equations: 𝑛+1𝑗=βˆ’1𝛾𝑗𝐡𝑗(0)=0,𝑛+1𝑗=βˆ’1𝛾𝑗𝐡𝑗(1)=0,𝑛+1𝑗=βˆ’1𝛿𝑗𝐡𝑗(0)=0,𝑛+1𝑗=βˆ’1𝛿𝑗𝐡𝑗(0)=0,(2.16) referring to the zero-boundary conditions: π‘’β„Ž(0)=0,π‘’β„Ž(1)=0,π‘£β„Ž(0)=0,π‘£β„Ž(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 𝑒(π‘₯) as well as for its second derivative 𝑣(π‘₯) 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 β€œqualocation” 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 𝐢 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,π‘βˆˆπΆ4𝐼,(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(π‘₯). It can be proved that the quadrature rule in (2.3) has an error bound of the form πΈβ„Ž||||𝒬(𝑔)=β„Žξ€œπ‘”||||(𝑔)βˆ’β‰€πΆπ‘›ξ“π‘–=1β„Ž4π‘˜β€–β€–π‘”(4)‖‖𝐿1(πΌπ‘˜).(4.2) This follows from Peano’s kernel theorem (see [11]).

The following inequality is frequently used in our analysis. If π‘£βˆˆπ‘Šπ‘šπ‘(𝐸) with π‘βˆˆ[1,∞], then there exists a positive constant 𝐢 depending only on π‘š such that, for any 𝛿satisfying 0<𝛿≀|𝐸|≀1,β€–π‘£β€–π‘Šπ‘–π‘(𝐸)ξ‚ƒπ›Ώβ‰€πΆπ‘šβˆ’π‘–β€–π‘£β€–π‘Šπ‘šπ‘(𝐸)+π›Ώβˆ’π‘–β€–π‘£β€–πΏπ‘(𝐸)ξ‚„,0β‰€π‘–β‰€π‘šβˆ’1,(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: πΏπ‘£βˆΆ=π‘£ξ…žξ…ž.(4.4) The adjoint operator πΏβˆ— with corresponding adjoint boundary condition is defined as follows: πΏβˆ—πœ™=πœ™ξ…žξ…ž,πœ™(0)=πœ™(1)=0.(4.5) Since 𝐿 is a self-adjoint operator, we mention below the regularity of πΏβˆ—(equal to 𝐿) in the π‘ž norm. We make a stronger assumption as in Sloan et al. [4] that for arbitrary π‘žβˆˆ[1,∞], there exists a positive constant 𝐢 such that β€–πΏβˆ—π‘’β€–πΏπ‘ž(𝐼)β‰₯πΆβ€–π‘’β€–π‘Š2π‘ž(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], β€–πœ™β€–πΏβˆž(πΌπ‘˜)β‰€β€–πœ™β€–π‘Š1𝑝(πΌπ‘˜);1β‰€π‘β‰€βˆž,πœ™βˆˆπ‘Š1π‘ξ€·πΌπ‘˜ξ€Έ.(4.7)

5. Convergence Analysis

Hereafter throughout this section, for 𝑝 and π‘ž with 1≀𝑝,π‘žβ‰€βˆž,s  and π‘βˆ’1+π‘žβˆ’1=1, we use the following notations: ‖𝑣‖0,𝑝=‖𝑣‖𝐿𝑝,‖𝑣‖𝑠,𝑝=β€–π‘£β€–π‘Šπ‘ π‘,‖𝑣‖𝑠,𝑝,π‘˜=β€–π‘£β€–π‘Šπ‘ π‘(πΌπ‘˜).(5.1) Let us denote the error between 𝑒 and π‘’β„Ž by πœ€β„Ž and the error between 𝑣 and π‘£β„Ž by π‘’β„Ž, respectively, that is, πœ€β„Ž=π‘’βˆ’π‘’β„Ž and π‘’β„Ž=π‘£βˆ’π‘£β„Ž. Using (2.11) and (1.5), we obtain the following error equations: ξ«π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž=ξ«π‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž=ξ«π‘£ξ…žξ…ž,πœ™β„Žξ¬β„Žβˆ’βŸ¨π‘“βˆ’π‘π‘’β„Ž,πœ™β„ŽβŸ©β„Žξ«π‘ξ€·=βˆ’π‘’βˆ’π‘’β„Žξ€Έ,πœ™β„Žξ¬β„Ž=βˆ’βŸ¨π‘πœ€β„Ž,πœ™β„ŽβŸ©β„Ž,(5.2) and therefore we get ξ«π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž=βˆ’βŸ¨π‘πœ€β„Ž,πœ™β„ŽβŸ©β„Ž,πœ™β„Žβˆˆπ‘†β„Ž,1.(5.3) Further, using (2.10) and (1.4),ξ«πœ€β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž=ξ«π‘’ξ…žξ…žβˆ’π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž=ξ«π›Όξ€·π‘£βˆ’π‘£β„Žξ€Έ,πœ™β„Žξ¬β„Ž=βŸ¨π›Όπ‘’β„Ž,πœ™β„ŽβŸ©β„Ž,(5.4) and therefore we have ξ«πœ€β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž=βŸ¨π›Όπ‘’β„Ž,πœ™β„ŽβŸ©β„Ž,πœ™β„Žβˆˆπ‘†β„Ž,1.(5.5) The following lemma gives estimates for the error in the quadrature rule for the term (π‘’β„Žξ…žξ…žπœ’β„Ž) and (πœ€β„Žξ…žξ…žπœ’β„Ž) for πœ’β„Žβˆˆπ‘†β„Ž,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 πœ’β„Žβˆˆπ‘†β„Ž,1 and h sufficiently small, (a)πΈβ„Ž(π‘’β„Žξ…žξ…žπœ’β„Ž)β‰€πΆβ„Ž4‖𝑣‖6,π‘β€–πœ’β„Žβ€–1,π‘ž, (b)πΈβ„Ž(π‘’β„Žξ…žξ…žπœ’β„Ž)β‰€πΆβ„Ž3‖𝑣‖6,π‘β€–πœ’β„Žβ€–0,π‘ž, (c)πΈβ„Ž(πœ€β„Žξ…žξ…žπœ’β„Ž)β‰€πΆβ„Ž4‖𝑒‖6,π‘β€–πœ’β„Žβ€–1,π‘ž, (d)πΈβ„Ž(πœ€β„Žξ…žξ…žπœ’β„Ž)β‰€πΆβ„Ž3‖𝑒‖6,π‘β€–πœ’β„Žβ€–0,π‘ž.

The following result gives estimate for πœ€β„Ž(π‘₯), where π‘₯ is any arbitrary point in 𝐼. This estimate is crucial for our error analysis.

Lemma 5.2. Let 𝑒 be the weak solution of (1.4) defined through (2.7). Further, let π‘’β„Ž be the corresponding discrete Petrov-Galerkin solution defined through (2.10). Then, the error πœ€β„Ž=π‘’βˆ’π‘’β„Ž satisfies ||πœ€β„Žξ€·π‘₯ξ€Έ||ξ€Ίβ„Žβ‰€πΆ2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,𝑝,(5.6) where π‘₯ is an arbitrary point in [0,1].

Proof. For a given π‘₯∈[0,1], let Ξ¦ be an element of 𝐿𝑝⋂(𝐼)𝐢(𝐼) satisfying the following auxiliary problem: Ξ¦ξ…žξ…žξ€½=0,π‘₯βˆˆπΌβˆ’π‘₯ξ€Ύ,Ξ¦(0)=Ξ¦(1)=0,Ξ¦β€²βˆ’ξ€·π‘₯ξ€Έβˆ’Ξ¦ξ…ž+ξ€·π‘₯ξ€Έ=βˆ’1.(5.7) The above problem has a solution. For example, ξ‚»ξ€·Ξ¦(π‘₯)=ξ€Έπ‘₯βˆ’1π‘₯,0≀π‘₯≀π‘₯,π‘₯(π‘₯βˆ’1),π‘₯≀π‘₯≀1(5.8) satisfies the above differential equation, the boundary conditions, and the jump condition.
Let us define Ξ¨ as follows: ΦΨ(π‘₯)=ξ…žξ…žξ€½,π‘₯βˆˆπΌβˆ’π‘₯ξ€Ύ,0,atπ‘₯=π‘₯.(5.9) Then, Ξ¨=0  a.e. on 𝐼. We first multiply πœ€β„Ž with Ξ¨ and then integrate over 𝐼. On applying integration by parts, using the fact that πœ€β„Ž(0)=πœ€β„Ž(1)=0 and the jump condition for Ξ¦β€², we obtain ξ€·πœ€0=β„Žξ€Έ=ξ€œ,Ξ¨π‘₯0πœ€β„Žξ€œΞ¨+1π‘₯πœ€β„Žξ€œΞ¨=π‘₯0πœ€β„ŽΞ¦ξ…žξ…ž+ξ€œ1π‘₯πœ€β„ŽΞ¦ξ…žξ…ž=ξ€Ίπœ€β„ŽΞ¦ξ…žξ€»π‘₯0βˆ’ξ€œπ‘₯0πœ€ξ…žβ„ŽΞ¦ξ…ž+ξ€Ίπœ€β„ŽΞ¦ξ…žξ€»1π‘₯βˆ’ξ€œ1π‘₯πœ€ξ…žβ„ŽΞ¦ξ…ž=πœ€β„Žξ€·π‘₯Ξ¦ξ€Έξ€Ίξ…žβˆ’ξ€·π‘₯ξ€Έβˆ’Ξ¦ξ…ž+ξ€·π‘₯βˆ’ξ€œξ€Έξ€»π‘₯0πœ€ξ…žβ„ŽΞ¦ξ…žβˆ’ξ€œ1π‘₯πœ€ξ…žβ„ŽΞ¦ξ…ž=βˆ’πœ€β„Žξ€·π‘₯ξ€Έβˆ’ξ€œπ‘₯0πœ€ξ…žβ„ŽΞ¦ξ…žβˆ’ξ€œ1π‘₯πœ€ξ…žβ„ŽΞ¦ξ…ž.(5.10) Applying integration by parts once again, using boundary condition for Ξ¦ and the continuity of Ξ¦, we obtain 0=βˆ’πœ€β„Žξ€·π‘₯ξ€Έβˆ’ξƒ―ξ€Ίπœ€ξ…žβ„ŽΞ¦ξ€»π‘₯0βˆ’ξ€œπ‘₯0πœ€β„Žξ…žξ…žξ€Ίπœ€Ξ¦+ξ…žβ„ŽΞ¦ξ€»1π‘₯βˆ’ξ€œ1π‘₯πœ€β„Žξ…žξ…žΞ¦ξƒ°=βˆ’πœ€β„Žξ€·π‘₯ξ€Έ+ξ€·πœ€β„Žξ…žξ…žξ€Έ,Ξ¦,(5.11) that is, πœ€β„Ž(π‘₯)=(πœ€β„Žξ…žξ…ž,Ξ¦). Let Ξ¦β„Ž be the linear interpolant of Ξ¦. Then, we have πœ€β„Žξ€·π‘₯ξ€Έ=ξ€·πœ€β„Žξ…žξ…ž,Ξ¦βˆ’Ξ¦β„Žξ€Έ+ξ€·πœ€β„Žξ…žξ…ž,Ξ¦β„Žξ€Έβˆ’ξ«πœ€β„Žξ…žξ…ž,Ξ¦β„Žξ¬β„Ž+ξ«πœ€β„Žξ…žξ…ž,Ξ¦β„Žξ¬β„Ž||πœ€β„Žξ€·π‘₯ξ€Έ||≀||ξ€·πœ€β„Žξ…žξ…ž,Ξ¦βˆ’Ξ¦β„Žξ€Έ||+||πΈβ„Žξ€·πœ€β„Žξ…žξ…žΞ¦β„Žξ€Έ||+||ξ«πœ€β„Žξ…žξ…ž,Ξ¦β„Žξ¬β„Ž||≀𝑇1+𝑇2+𝑇3.(5.12) We know that β€–β€–Ξ¦β„Žβ€–β€–1,π‘žβ‰€β€–β€–Ξ¦βˆ’Ξ¦β„Žβ€–β€–1,π‘ž+β€–Ξ¦β€–1,π‘žβ‰€πΆβ„Žβ€–Ξ¦β€–2,π‘ž+β€–Ξ¦β€–2,π‘žβ‰€πΆβ€–Ξ¦β€–2,π‘ž.(5.13) We now compute the estimates for the terms 𝑇1, 𝑇2, and 𝑇3 as follows: 𝑇1=||ξ€·πœ€β„Žξ…žξ…ž,Ξ¦βˆ’Ξ¦β„Žξ€Έ||β‰€β€–β€–πœ€β„Žξ…žξ…žβ€–β€–0,π‘β€–β€–Ξ¦βˆ’Ξ¦β„Žβ€–β€–0,π‘žβ‰€πΆβ„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝‖Φ‖2,π‘ž.(5.14) Using Lemma 5.1(c) and (5.13), we obtain 𝑇2=||πΈβ„Žξ€·πœ€β„Žξ…žξ…žΞ¦β„Žξ€Έ||β‰€πΆβ„Ž4‖𝑒‖6,𝑝‖Φ‖2,π‘ž.(5.15) Using (5.5), (2.3), and the Sobolev embedding theorem (4.7) locally on πΌπ‘˜ for both β€–π‘’β„Žβ€–0,∞,π‘˜ and β€–Ξ¦β„Žβ€–0,∞,π‘˜, we have 𝑇3=||ξ«πœ€β„Žξ…žξ…ž,Ξ¦β„Žξ¬β„Ž||=||βŸ¨π›Όπ‘’β„Ž,Ξ¦β„ŽβŸ©β„Ž||β‰€πΆπ‘›ξ“π‘˜=1β„Žπ‘˜2β€–β€–π‘’β„Žβ€–β€–0,∞,π‘˜β€–β€–Ξ¦β„Žβ€–β€–0,∞,π‘˜β‰€πΆπ‘›ξ“π‘˜=1β„Žπ‘˜2β€–β€–π‘’β„Žβ€–β€–1,𝑝,π‘˜β€–β€–Ξ¦β„Žβ€–β€–1,π‘ž,π‘˜.(5.16) Using HΓΆlder's inequality for sums and (5.13), we have 𝑇3β€–β€–π‘’β‰€πΆβ„Žβ„Žβ€–β€–1,π‘β€–β€–Ξ¦β„Žβ€–β€–1,π‘žβ€–β€–π‘’β‰€πΆβ„Žβ„Žβ€–β€–1,𝑝‖Φ‖2,π‘ž.(5.17) For Ξ¦ satisfying the auxiliary problem, it is easy to verify that β€–Ξ¦β€–2,π‘žβ‰€πΎ, where 𝐾 is a constant not depending on β„Ž.
Using 𝑇1, 𝑇2, and 𝑇3 in (5.12), we have ||πœ€β„Žξ€·π‘₯ξ€Έ||ξ€Ίβ„Žβ‰€πΆ2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,𝑝.(5.18) This completes the proof.

In the following lemma, we initially compute the error (π‘£βˆ’π‘£β„Ž) in terms of (π‘’βˆ’π‘’β„Ž), and then later on we establish an optimal estimate of error (π‘£βˆ’π‘£β„Ž) independent of (π‘’βˆ’π‘’β„Ž).

Lemma 5.3. Let 𝑒 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 π‘’β„Ž and π‘£β„Ž be the corresponding discrete Petrov-Galerkin solutions defined through (2.10) and (2.11), respectively. Then the estimates of the errors π‘’β„Ž=π‘£βˆ’π‘£β„Ž in 𝐿𝑝, π‘Š1𝑝, and π‘Š2𝑝 norms are given as follows: β€–β€–π‘’β„Žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ4‖𝑣‖6,𝑝+β„Ž5‖𝑒‖6,𝑝+β„Ž3β€–β€–πœ€β„Žβ€–β€–2,𝑝,β€–β€–π‘’β„Žβ€–β€–1,π‘ξ€Ίβ„Žβ‰€πΆ3‖𝑣‖6,𝑝+β„Ž4‖𝑒‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝,β€–β€–π‘’β„Žβ€–β€–2,π‘ξ€Ίβ„Žβ‰€πΆ2‖𝑣‖6,𝑝+β„Ž4‖𝑒‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝.(5.19)

Proof. Let πœ‚ be an arbitrary element of πΏπ‘ž, and let πœ™βˆˆπ‘Š2π‘ž be the solution of the auxiliary problem πΏβˆ—πœ™=πœ‚,πœ™(0)=πœ™(1)=0.(5.20) We now have ξ€·π‘’β„Žξ€Έ=𝑒,πœ‚β„Ž,πΏβˆ—πœ™ξ€Έ=ξ€·πΏπ‘’β„Žξ€Έ=𝑒,πœ™β„Žξ…žξ…ž,πœ™βˆ’πœ™β„Žξ€Έ+ξ€·π‘’β„Žξ…žξ…ž,πœ™β„Žξ€Έ=ξ€·π‘’β„Žξ…žξ…ž,πœ™βˆ’πœ™β„Žξ€Έ+ξ€·π‘’β„Žξ…žξ…ž,πœ™β„Žξ€Έβˆ’ξ«π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž+ξ«π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž=ξ€·π‘’β„Žξ…žξ…ž,πœ™βˆ’πœ™β„Žξ€Έ+πΈβ„Žξ€·π‘’β„Žξ…žξ…žπœ™β„Žξ€Έ+ξ«π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž,||ξ€·π‘’β„Žξ€Έ||≀||𝑒,πœ‚β„Žξ…žξ…ž,πœ™βˆ’πœ™β„Žξ€Έ||+||πΈβ„Žξ€·π‘’β„Žξ…žξ…žπœ™β„Žξ€Έ||+||ξ«π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž||≀𝑇4+𝑇5+𝑇6,(5.21) where πœ™β„Žβˆˆπ‘†β„Ž,1 is the linear interpolant of πœ™.
We know that β€–β€–πœ™β„Žβ€–β€–1,π‘žβ‰€β€–β€–πœ™βˆ’πœ™β„Žβ€–β€–1,π‘ž+β€–πœ™β€–1,π‘žβ‰€πΆβ„Žβ€–πœ™β€–2,π‘ž+β€–πœ™β€–2,π‘žβ‰€πΆβ€–πœ™β€–2,π‘ž.(5.22) We shall compute the estimates for the terms 𝑇4, 𝑇5, and 𝑇6 as follows: 𝑇4=||ξ€·π‘’β„Žξ…žξ…ž,πœ™βˆ’πœ™β„Žξ€Έ||β‰€β€–β€–π‘’β„Žξ…žξ…žβ€–β€–0,π‘β€–β€–πœ™βˆ’πœ™β„Žβ€–β€–0,π‘žβ‰€πΆβ„Ž2β€–β€–π‘’β„Žβ€–β€–2,π‘β€–πœ™β€–2,π‘ž,𝑇5=||πΈβ„Žξ€·π‘’β„Žξ…žξ…žπœ™β„Žξ€Έ||β‰€πΆβ„Ž4‖𝑣‖6,π‘β€–β€–πœ™β„Žβ€–β€–1,π‘žβ‰€πΆβ„Ž4‖𝑣‖6,π‘β€–πœ™β€–2,π‘žbyLemma5.1(a),(5.22).(5.23) Using (5.3), (2.3), and the Sobolev embedding theorem (4.7) locally on πΌπ‘˜ for β€–πœ™β„Žβ€–0,∞,π‘˜, we have 𝑇6=||ξ«π‘’β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž||=||βˆ’βŸ¨π‘πœ€β„Ž,πœ™β„ŽβŸ©β„Ž||β‰€πΆπ‘›ξ“π‘˜=1β„Žπ‘˜2β€–β€–πœ€β„Žβ€–β€–0,∞,π‘˜β€–β€–πœ™β„Žβ€–β€–0,∞,π‘˜β‰€πΆπ‘›ξ“π‘˜=1β„Žπ‘˜2β€–β€–πœ€β„Žβ€–β€–0,∞,π‘˜β€–β€–πœ™β„Žβ€–β€–1,π‘ž,π‘˜.(5.24) Using HΓΆlder's inequality for sums, Lemma 5.2, and (5.22), we obtain 𝑇6ξ€Ίβ„Žβ‰€πΆβ„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,π‘ξ€»β€–β€–πœ™β„Žβ€–β€–1,π‘žξ€Ίβ„Žβ‰€πΆ3β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž5‖𝑒‖6,𝑝+β„Ž2β€–β€–π‘’β„Žβ€–β€–1,π‘ξ€»β€–πœ™β€–2,π‘ž.(5.25) Substituting 𝑇4, 𝑇5, and 𝑇6 in (5.21), we have ||ξ€·π‘’β„Žξ€Έ||ξ€Ίβ„Ž,πœ‚β‰€πΆ2β€–β€–π‘’β„Žβ€–β€–2,𝑝+β„Ž4‖𝑣‖6,𝑝+β„Ž3β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž5‖𝑒‖6,𝑝+β„Ž2β€–β€–π‘’β„Žβ€–β€–1,π‘ξ€»β€–πœ™β€–2,π‘ž.(5.26) Using (4.6) and the regularity of the auxiliary problem, we have β€–πœ™β€–2,π‘žβ‰€πΆβ€–πœ‚β€–0,π‘ž. Since πœ‚βˆˆπΏπ‘ž is arbitrary, we have β€–β€–π‘’β„Žβ€–β€–0,π‘ξ€·β„Žβ‰€πΆ2β€–β€–π‘’β„Žβ€–β€–2,𝑝+β„Ž3β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑣‖6,𝑝+β„Ž5‖𝑒‖6,𝑝.(5.27) We now estimate β€–π‘’β„Žξ…žξ…žβ€– via a projection argument. Let π‘ƒβ„Ž be the orthogonal projection onto π‘†β„Ž,1 with respect to 𝐿2 inner product defined by ξ€·π‘£ξ…žξ…žβˆ’π‘ƒβ„Žπ‘£ξ…žξ…ž,πœ“β„Žξ€Έ=0,πœ“β„Žβˆˆπ‘†β„Ž,1.(5.28) The domain of π‘ƒβ„Ž may be taken to be 𝐿1. From Crouzeix and ThomΓ©e [13] and de Boor [14], it is seen that the 𝐿2 projection is stable. Thus, β€–β€–π‘ƒβ„Žπ‘£β€–β€–0,𝑝≀𝐢‖𝑣‖0,𝑝.(5.29) Then the error π‘’β„Žξ…žξ…ž can be interpreted in terms of the error of the above projection: β€–β€–π‘’β„Žξ…žξ…žβ€–β€–0,𝑝=β€–β€–π‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…žβ€–β€–0,π‘β‰€β€–β€–π‘£ξ…žξ…žβˆ’π‘ƒβ„Žπ‘£ξ…žξ…žβ€–β€–0,𝑝+β€–β€–π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…žβ€–β€–0,𝑝.(5.30) From the stability property (5.29), the error in the projection follows as in de Boor [14], that is, β€–β€–π‘£ξ…žξ…žβˆ’π‘ƒβ„Žπ‘£ξ…žξ…žβ€–β€–0,π‘β‰€πΆβ„Ž2β€–β€–π‘£ξ…žξ…žβ€–β€–2,π‘β‰€πΆβ„Ž2‖𝑣‖4,𝑝.(5.31) Then the remaining task is to compute the estimate of β€–π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…žβ€–0,𝑝.
For πœ“β„Žβˆˆπ‘†β„Ž,1, ξ€·π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ“β„Žξ€Έ=ξ€·π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£ξ…žξ…ž+π‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ“β„Žξ€Έ=ξ€·π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£ξ…žξ…ž,πœ“β„Žξ€Έ+ξ€·π‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ“β„Žξ€Έ=ξ€·π‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ“β„Žξ€Έξ€·π‘ƒusing(5.28),β„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ“β„Žξ€Έ=ξ€·π‘’β„Žξ…žξ…ž,πœ“β„Žξ€Έ=ξ€·π‘’β„Žξ…žξ…ž,πœ“β„Žξ€Έβˆ’ξ«π‘’β„Žξ…žξ…ž,πœ“β„Žξ¬β„Ž+ξ«π‘’β„Žξ…žξ…ž,πœ“β„Žξ¬β„Ž=πΈβ„Žξ€·π‘’β„Žξ…žξ…žπœ“β„Žξ€Έ+ξ«π‘’β„Žξ…žξ…ž,πœ“β„Žξ¬β„Ž,||ξ€·π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ“β„Žξ€Έ||≀||πΈβ„Žξ€·π‘’β„Žξ…žξ…žπœ“β„Žξ€Έ||+||ξ«π‘’β„Žξ…žξ…ž,πœ“β„Žξ¬β„Ž||≀𝑇7+𝑇8.(5.32) We shall compute the estimates for the terms 𝑇7 and 𝑇8𝑇7=||πΈβ„Žξ€·π‘’β„Žξ…žξ…žπœ“β„Žξ€Έ||β‰€πΆβ„Ž3‖𝑣‖6,π‘β€–β€–πœ“β„Žβ€–β€–0,π‘ž(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=||ξ«π‘’β„Žξ…žξ…ž,πœ“β„Žξ¬β„Ž||ξ€Ίβ„Žβ‰€πΆ2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,π‘ξ€»β€–β€–πœ“β„Žβ€–β€–0,π‘ž,(5.34) where we have used the inverse inequality β€–πœ“β„Žβ€–1,π‘ž,π‘˜β‰€β„Žπ‘˜βˆ’1β€–πœ“β„Žβ€–0,π‘ž,π‘˜ locally. Using 𝑇7 and 𝑇8 in (5.32), we get ||ξ€·π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ“β„Žξ€Έ||ξ€Ίβ„Žβ‰€πΆ3‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,π‘ξ€»β€–β€–πœ“β„Žβ€–β€–0,π‘ž.(5.35) We now show the above inequality for πœ‚βˆˆπΏπ‘ž to obtain β€–π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…žβ€–0,𝑝.
Now let πœ‚ be an arbitrary element of πΏπ‘ž. Then since π‘£β„Žξ…žξ…žβˆˆπ‘†β„Ž,1, it follows from the definition of π‘ƒβ„Žπœ‚, (5.35), and (5.29) with 𝑝 replaced by π‘ž, that 𝑃0=β„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,πœ‚βˆ’π‘ƒβ„Žπœ‚ξ€Έ,||ξ€·π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…žξ€Έ||=||𝑃,πœ‚β„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…ž,π‘ƒβ„Žπœ‚ξ€Έ||ξ€Ίβ„Žβ‰€πΆ3‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,π‘ξ€»β€–β€–π‘ƒβ„Žπœ‚β€–β€–0,π‘žξ€Ίβ„Žβ‰€πΆ3‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,π‘ξ€»β€–πœ‚β€–0,π‘ž,β€–β€–π‘ƒβ„Žπ‘£ξ…žξ…žβˆ’π‘£β„Žξ…žξ…žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ3‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,𝑝.(5.36) Now, from (5.30), (5.31), and (5.36), we conclude that β€–β€–π‘’β„Žξ…žξ…žβ€–β€–0,π‘β‰€πΆβ„Ž2‖𝑣‖4,π‘ξ€Ίβ„Ž+𝐢3‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,π‘ξ€»ξ€Ίβ„Žβ‰€πΆ2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,𝑝.(5.37) Now, using the fact β€–π‘’β„Žβ€–2,π‘β‰€β€–π‘’β„Žβ€–1,𝑝+β€–π‘’β„Žξ…žξ…žβ€–0,𝑝 and the above estimate, we have β€–β€–π‘’β„Žβ€–β€–2,π‘β‰€β€–β€–π‘’β„Žβ€–β€–1,π‘ξ€Ίβ„Ž+𝐢2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,π‘ξ€»ξ€Ίβ€–β€–π‘’β‰€πΆβ„Žβ€–β€–1,𝑝+β„Ž2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,π‘ξ€»ξ€Ίβ€–β€–π‘’β‰€πΆβ„Žβ€–β€–1,𝑝+β„Ž2‖𝑣‖6,𝑝+β„Ž2β€–πœ€β„Žβ€–2,𝑝+β„Ž4‖𝑒‖6,𝑝.(5.38) Now using (4.3) with π‘š=2 and 𝑖=1, we have β€–β€–π‘’β„Žβ€–β€–1,π‘ξ€·β„Žβ‰€πΆβˆ’1β€–β€–π‘’β„Žβ€–β€–0,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–2,𝑝.(5.39) Substituting (5.39) in the above expression, we obtain β€–β€–π‘’β„Žβ€–β€–2,π‘β„Žβ‰€πΆξ€Ίξ€·βˆ’1β€–β€–π‘’β„Žβ€–β€–0,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–2,𝑝+β„Ž2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝.(5.40) For sufficiently small β„Ž, we have β€–β€–π‘’β„Žβ€–β€–2,π‘ξ€Ίβ„Žβ‰€πΆβˆ’1β€–β€–π‘’β„Žβ€–β€–0,𝑝+β„Ž2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝.(5.41) Using (5.41) in (5.27), β€–β€–π‘’β„Žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ2ξ€·β„Žβˆ’1β€–β€–π‘’β„Žβ€–β€–0,𝑝+β„Ž2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝+β„Ž4‖𝑣‖6,𝑝+β„Ž5‖𝑒‖6,𝑝+β„Ž3β€–β€–πœ€β„Žβ€–β€–2,𝑝.(5.42) For sufficiently small β„Ž, we get β€–β€–π‘’β„Žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ4‖𝑣‖6,𝑝+β„Ž5‖𝑒‖6,𝑝+β„Ž3β€–β€–πœ€β„Žβ€–β€–2,𝑝.(5.43) Using (5.43) in (5.41), we have β€–β€–π‘’β„Žβ€–β€–2,π‘ξ€Ίβ„Žβ‰€πΆβˆ’1ξ€·β„Ž4‖𝑣‖6,𝑝+β„Ž5‖𝑒‖6,𝑝+β„Ž3β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,π‘ξ€»ξ€Ίβ„Žβ‰€πΆ2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝.(5.44) Using (5.43) and (5.44) in (5.39), we have β€–β€–π‘’β„Žβ€–β€–1,π‘ξ€Ίβ„Žβ‰€πΆβˆ’1ξ€·β„Ž4‖𝑣‖6,𝑝+β„Ž5‖𝑒‖6,𝑝+β„Ž3β€–β€–πœ€β„Žβ€–β€–2,π‘ξ€Έξ€·β„Ž+β„Ž2‖𝑣‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,π‘ξ€Ίβ„Žξ€Έξ€»β‰€πΆ3‖𝑣‖6,𝑝+β„Ž4‖𝑒‖6,𝑝+β„Ž2β€–β€–πœ€β„Žβ€–β€–2,𝑝.(5.45) Equations (5.43), (5.44), and (5.45) give the required result.

We now compute the error estimate of πœ€β„Ž in 𝐿𝑝,π‘Š1𝑝,and π‘Š2𝑝 norms as has been done in the previous case.

Lemma 5.4. Let 𝑒 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 π‘’β„Ž and π‘£β„Ž be the corresponding discrete Petrov-Galerkin solutions defined through (2.10) and (2.11), respectively. Then the estimates of the errors πœ€β„Ž=π‘’βˆ’π‘’β„Ž in 𝐿𝑝,π‘Š1𝑝and π‘Š2𝑝 norms are given as follows: β€–β€–πœ€β„Žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,𝑝,β€–β€–πœ€β„Žβ€–β€–1,π‘ξ€Ίβ„Žβ‰€πΆ3‖𝑒‖6,𝑝+β€–β€–π‘’β„Žβ€–β€–1,𝑝,β€–β€–πœ€β„Žβ€–β€–2,π‘ξ€Ίβ„Žβ‰€πΆ2‖𝑒‖6,𝑝+β€–β€–π‘’β„Žβ€–β€–1,𝑝.(5.46)

Proof. Let 𝜌 be an arbitrary element of πΏπ‘ž, and let πœ™βˆˆπ‘Š2π‘ž be the unique solution of the auxiliary problem πΏβˆ—πœ™=𝜌,πœ™(0)=πœ™(1)=0.(5.47) Then we have ξ€·πœ€β„Žξ€Έ=ξ€·πœ€,πœŒβ„Ž,πΏβˆ—πœ™ξ€Έ=ξ€·πΏπœ€β„Žξ€Έ=ξ€·πœ€,πœ™β„Žξ…žξ…žξ€Έ=ξ€·πœ€,πœ™β„Žξ…žξ…ž,πœ™βˆ’πœ™β„Žξ€Έ+ξ€·πœ€β„Žξ…žξ…ž,πœ™β„Žξ€Έβˆ’ξ«πœ€β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž+ξ«πœ€β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž,(5.48) where πœ™β„Žβˆˆπ‘†β„Ž,1 is a linear interpolant of πœ™, ||ξ€·πœ€β„Ž,πœŒξ€Έ||≀||ξ€·πœ€β„Žξ…žξ…ž,πœ™βˆ’πœ™β„Žξ€Έ||+||πΈβ„Žξ€·πœ€β„Žξ…žξ…žπœ™β„Žξ€Έ||+||ξ«πœ€β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž||≀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: 𝑇9β‰€πΆβ„Ž2β€–β€–πœ€β„Žβ€–β€–2,π‘β€–πœ™β€–2,π‘ž,𝑇10β‰€πΆβ„Ž4‖𝑒‖6,π‘β€–πœ™β€–2,π‘ž,(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 πΌπ‘˜ for β€–πœ™β„Žβ€–0,∞,π‘˜ and β€–π‘’β„Žβ€–0,∞,π‘˜ to estimate 𝑇11, we have 𝑇11=||ξ«πœ€β„Žξ…žξ…ž,πœ™β„Žξ¬β„Ž||=||βŸ¨π›Όπ‘’β„Ž,πœ™β„ŽβŸ©β„Ž||β‰€πΆπ‘›ξ“π‘˜=1β„Žπ‘˜2β€–β€–π‘’β„Žβ€–β€–0,∞,π‘˜β€–β€–πœ™β„Žβ€–β€–0,∞,π‘˜β‰€πΆπ‘›ξ“π‘˜=1β„Žπ‘˜2β€–β€–π‘’β„Žβ€–β€–0,∞,π‘˜β€–β€–πœ™β„Žβ€–β€–1,π‘ž,π‘˜β‰€πΆπ‘›ξ“π‘˜=1β„Žπ‘˜2β€–β€–π‘’β„Žβ€–β€–1,𝑝,π‘˜β€–β€–πœ™β„Žβ€–β€–1,π‘ž,π‘˜.(5.51) Further, using HΓΆlder's inequality for sums and (5.22), we obtain 𝑇11β€–β€–π‘’β‰€πΆβ„Žβ„Žβ€–β€–1,π‘β€–β€–πœ™β„Žβ€–β€–1,π‘žβ€–β€–π‘’β‰€πΆβ„Žβ„Žβ€–β€–1,π‘β€–πœ™β€–2,π‘ž.(5.52) Substituting the estimates 𝑇9, 𝑇10, and 𝑇11 in (5.49), we obtain ||ξ€·πœ€β„Ž,πœŒξ€Έ||ξ€Ίβ„Žβ‰€πΆ2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,π‘ξ€»β€–πœ™β€–2,π‘ž.(5.53) Using (4.6) and regularity of the auxiliary problem, we have β€–πœ™β€–2,π‘žβ‰€πΆβ€–πœŒβ€–π‘œ,π‘ž. Since πœŒβˆˆπΏπ‘ž is arbitrary, we have β€–β€–πœ€β„Žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ2β€–β€–πœ€β„Žβ€–β€–2,𝑝+β„Ž4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,𝑝.(5.54) The estimate of β€–πœ€β„Žξ…žξ…žβ€–0,𝑝 can be obtained through a projection argument as mentioned in Lemma 5.3 as β€–β€–πœ€β„Žξ…žξ…žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ2‖𝑒‖6,𝑝+β€–β€–π‘’β„Žβ€–β€–1,𝑝,(5.55) where we have used Lemma 5.1(d). In a similar manner we can compute the estimates for β€–πœ€β„Žβ€–0,𝑝, β€–πœ€β„Žβ€–1,𝑝 and β€–πœ€β„Žβ€–2,𝑝 as β€–β€–πœ€β„Žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ4‖𝑒‖6,𝑝‖‖𝑒+β„Žβ„Žβ€–β€–1,𝑝,β€–β€–πœ€β„Žβ€–β€–1,π‘ξ€Ίβ„Žβ‰€πΆ3‖𝑒‖6,𝑝+β€–β€–π‘’β„Žβ€–β€–1,𝑝,β€–β€–πœ€β„Žβ€–β€–2,π‘ξ€Ίβ„Žβ‰€πΆ2‖𝑒‖6,𝑝+β€–β€–π‘’β„Žβ€–β€–1,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 𝑒 and 𝑣 satisfy (1.4) and (1.5), respectively, with (4.1). Assume also that π‘’βˆˆπ‘Š6𝑝 and π‘£βˆˆπ‘Š6𝑝, where π‘βˆˆ[1,∞]. Then (2.10) and (2.11) have unique solutions π‘’β„Žβˆˆ0π‘†β„Ž,3 and π‘£β„Žβˆˆ0π‘†β„Ž,3, respectively, and for β„Ž sufficiently small, one has β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–π‘–,π‘β‰€πΆβ„Ž4βˆ’π‘–ξ€Ίβ€–π‘’β€–6,𝑝+‖𝑣‖6,𝑝,β€–β€–π‘£βˆ’π‘£β„Žβ€–β€–π‘–,π‘β‰€πΆβ„Ž4βˆ’π‘–ξ€Ίβ€–π‘’β€–6,𝑝+‖𝑣‖6,𝑝,𝑖=0,1,2.(5.57)

Proof. Assume temporarily that solutions π‘’β„Ž and π‘£β„Ž of (2.10) and (2.11), respectively, exist. Using (5.46) in (5.45), we obtain β€–β€–π‘’β„Žβ€–β€–1,π‘ξ€Ίβ„Žβ‰€πΆ3‖𝑣‖6,𝑝+β„Ž4‖𝑒‖6,𝑝+β„Ž2ξ€·β„Ž2‖𝑒‖6,𝑝+β€–π‘’β„Žβ€–1,𝑝.(5.58) For sufficiently small β„Ž, we have β€–β€–π‘’β„Žβ€–β€–1,π‘ξ€·β„Žβ‰€πΆ3‖𝑣‖6,𝑝+β„Ž4‖𝑒‖6,𝑝.(5.59) An application of the above in (5.46), we get β€–β€–πœ€β„Žβ€–β€–2,π‘ξ€Ίβ„Žβ‰€πΆ2‖𝑒‖6,𝑝+β„Ž3‖𝑣‖6,𝑝.(5.60) Apply (5.59) in (5.56) to have β€–β€–πœ€β„Žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ4‖𝑒‖6,𝑝+β„Ž4‖𝑣‖6,𝑝.(5.61) Use (5.60) in (5.43) to get β€–β€–π‘’β„Žβ€–β€–0,π‘ξ€Ίβ„Žβ‰€πΆ4‖𝑣‖6,𝑝+β„Ž5‖𝑒‖6,𝑝.(5.62) Using (5.60) in (5.44), we obtain β€–β€–π‘’β„Žβ€–β€–2,π‘ξ€Ίβ„Žβ‰€πΆ2‖𝑣‖6,𝑝+β„Ž4‖𝑒‖6,𝑝.(5.63) Using (5.61) and (5.60) in (5.39) with π‘’β„Ž replaced by πœ€β„Ž, we have β€–β€–πœ€β„Žβ€–β€–1,π‘ξ€Ίβ„Žβ‰€πΆ3‖𝑒‖6,𝑝+β„Ž3‖𝑣‖6,𝑝.(5.64) The required result can be obtained from estimates (5.59) to (5.64).

So far we have assumed temporarily that solutions π‘’β„Ž and π‘£β„Ž 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 π‘’β„Ž and π‘£β„Ž is square, existence of π‘’β„Žβˆˆ0π‘†β„Ž,3 and π‘£β„Žβˆˆ0π‘†β„Ž,3 for any π‘“βˆˆπΆ0(𝐼) will follow from uniqueness, that is, from the property that the corresponding homogeneous equations have only trivial solutions.

Suppose that π‘’β„Ž and π‘£β„Ž corresponding to 𝑒 and 𝑣 satisfy ξ«π‘’β„Žξ…žξ…žβˆ’π›Όπ‘£β„Ž,πœ’β„Žξ¬ξ«π‘£=0,β„Žξ…žξ…ž+π‘π‘’β„Ž,πœ’β„Žξ¬=0,πœ’β„Žβˆˆπ‘†β„Ž,1.(5.65) It follows from (5.61) and (5.62) (with 𝑒 replaced by 0 and eventually 𝑣≑0) that, for sufficiently small β„Ž, β€–β€–π‘’β„Žβ€–β€–0,𝑝‖‖𝑣≀0,β„Žβ€–β€–0,𝑝≀0,(5.66) and hence π‘’β„Žβ‰‘0 and π‘£β„Žβ‰‘0. Thus, uniqueness is proved, and hence existence follows from uniqueness.


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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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