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: 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.