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

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: 𝑑²𝑑𝑥²𝑎(𝑥)𝑑²𝑢𝑑𝑥²+𝑏(𝑥)𝑢=𝑓(𝑥),𝑥∈𝐼=(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,𝑥∈𝐼. 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,𝑥∈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: 𝑣𝑢^=𝛼𝑣,𝑥∈𝐼,with𝑢(0)=𝑢(1)=0,(1.4)_+𝑏𝑢=𝑓,𝑥∈𝐼,with𝑣(0)=𝑣(1)=0.(1.5) In this paper, the error analysis will take place in the usual Sobolev space 𝑊^𝑚𝑝(𝐼) defined on the domain 𝐼=(0,1) with 𝐻^𝑚(𝐼) denoting 𝑊^𝑚2(𝐼). The Sobolev norms are given below. For an open interval 𝐸 and a non negative integer 𝑚, ‖𝑣‖_𝑊^𝑚𝑝_(𝐸)=^𝑚_𝑖=0‖‖𝑣^(𝑖)‖‖^𝑝𝐿_𝑝(𝐸)^1/𝑝=max_{1≤𝑖≤𝑛},if1≤𝑝<∞,‖‖𝑣^(𝑖)‖‖_{𝐿∞(𝐸)},if𝑝=∞.(1.6) We suppress the dependence of the norms on 𝐼 when 𝐸=𝐼. Further, 𝐻^𝑚0(𝐼) denotes the function space {𝜙∈𝐻^𝑚(𝐼)∶𝜙(0)=𝜙(1)=0}.(1.7)

2. Continuous and Discrete 𝐻¹-Galerkin Formulation

Given 𝑛>1, let Π_𝑛∶0=𝑥₀<𝑥₁<⋯<𝑥_𝑛=1(2.1) be an arbitrary partition of [0,1] with the property that ℎ→0 as 𝑛→∞, where ℎ=max_{1≤𝑘≤𝑛}ℎ_𝑘 and ℎ_𝑘=𝑥_𝑘−𝑥_𝑘−1,𝑘=1,…,𝑛. Let (𝑢,𝑣) represent the 𝐿₂ inner product, and let ⟨𝑢,𝑣⟩_ℎ represent the discrete inner product of any two functions 𝑢,𝑣∈𝐿₂(𝐼) and be defined as follows: (𝑢,𝑣)=𝑢𝑣𝑑𝑥,⟨𝑢,𝑣⟩_ℎ=𝒬_ℎ(𝑢𝑣),(2.2) where 𝒬_ℎ is the fourth-order Gaussian quadrature rule: 𝒬_ℎ(𝑔)∶=12^𝑛_𝑖=1ℎ_𝑘𝑔𝑥_𝑘,1+𝑔𝑥_𝑘,2.(2.3) Here, 𝑥_𝑘,𝑖=𝑥_𝑘−1+𝜉_𝑖ℎ_𝑘,𝑖=1,2, are the two Gaussian points in the subinterval [𝑥_𝑘−1,𝑥_𝑘] with 𝜉₁=(1/2)(1−1/√3), 𝜉₂=1−𝜉₁.

Let us now consider the following cubic spline space as trial space: 𝑆_ℎ,3=𝜑∈𝐶²(𝐼)∶𝜑|_𝐼𝑘∈𝑃₃𝐼_𝑘,𝑘=1,2,…,𝑛,(2.4) where 𝑃_𝑟(𝐼_𝑘) is the space of polynomials of degree 𝑟 defined over the 𝑘th subinterval 𝐼_𝑘=[𝑥_𝑘−1,𝑥_𝑘].

The corresponding space with zero Dirichlet boundary condition is denoted by 𝑆⁰_ℎ,3=𝜑∈𝑆_ℎ,3∶𝜑(0)=𝜑(1)=0.(2.5) Further, let us consider the following piecewise linear space 𝑆_ℎ,1=𝜑∈𝐶(𝐼)∶𝜑|_𝐼𝑘∈𝑃₁𝐼_𝑘,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 𝑣𝑢^,𝜙=(𝛼𝑣,𝜙),𝜙∈𝐻²(0,1),(2.7)_+𝑏𝑢,𝜙=(𝑓,𝜙),𝜙∈𝐻²(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_ℎ,𝑣_ℎ}∈𝑆⁰_ℎ,3 such that 𝑣_ℎ𝑢^ℎ,𝜙_ℎ=𝛼𝑣_ℎ,𝜙_ℎ,𝜙_ℎ∈𝑆_ℎ,1,+𝑏𝑢_ℎ,𝜙_ℎ=𝑓,𝜙_ℎ,𝜙_ℎ∈𝑆_ℎ,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_ℎ,𝑣_ℎ}∈𝑆⁰_ℎ,3 such that 𝑣𝑢^ℎ,𝜙_ℎ_ℎ=⟨𝛼𝑣_ℎ,𝜙_ℎ⟩_ℎ,𝜙_ℎ∈𝑆_ℎ,1,(2.10)_ℎ+𝑏𝑢_ℎ,𝜙_ℎ_ℎ=⟨𝑓,𝜙_ℎ⟩_ℎ,𝜙_ℎ∈𝑆_ℎ,1.(2.11) The approximate solutions 𝑢_ℎ and 𝑣_ℎ without any conditions on boundary points are expressed as a linear combination of the B-splines as follows: 𝑢_ℎ(𝑥)=^𝑛+1_𝑗₌₋₁𝛾_𝑗𝐵_𝑗(𝑥),𝑣_ℎ(𝑥)=^𝑛+1_𝑗₌₋₁𝛿_𝑗𝐵_𝑗(𝑥),(2.12) where the 𝑗th basis 𝐵_𝑗(𝑥) of the cubic B-splines space 𝑆_ℎ,3 for 𝑗=−1,0,1,2,…,𝑛,𝑛+1 is given below: 𝐵_𝑗(𝑥)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩0,if𝑥≤𝑥^𝑗−21,6ℎ³1𝑥−𝑥_𝑗−2³,if𝑥_𝑗−2≤𝑥≤𝑥_𝑗−1,6ℎ³ℎ³1+3ℎ²𝑥−𝑥_𝑗−1+3ℎ𝑥−𝑥_𝑗−1²−3𝑥−𝑥_𝑗−1³,if𝑥^𝑗−1≤𝑥≤𝑥_𝑗,6ℎ³ℎ³1+3ℎ²𝑥_𝑗+1−𝑥+3ℎ𝑥_𝑗+1−𝑥²−3𝑥_𝑗+1−𝑥³,if𝑥^𝑗≤𝑥≤𝑥_𝑗+1,6ℎ³0,𝑥_𝑗+2−𝑥³,iiff𝑥𝑥≥_𝑗+1𝑥≤𝑥_𝑗+2≤𝑥_𝑗+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 𝑥₋₃,𝑥₋₂,𝑥₋₁ on the left-hand side and 𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+3 on the right-hand side, respectively. Further, the 𝑖th basis 𝜙_𝑖(𝑥) of the piecewise linear “hat” splines space 𝑆_ℎ,1 for 𝑖=0,1,2,…,𝑛 is given below: 𝜙_𝑖(𝑥)=⎧⎪⎪⎨⎪⎪⎩0,if𝑥≤𝑥^𝑖−11,ℎ𝑥−𝑥^𝑖−1,if𝑥_𝑖−11≤𝑥≤𝑥_𝑖,ℎ𝑥^𝑖+10,−𝑥,iiff𝑥𝑥≥_𝑖≤𝑥𝑥_𝑖+1≤𝑥_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 𝑥₋₁ on the left-hand side and 𝑥_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=−1𝛾_𝑗𝐵^𝑗,𝜙_𝑖_ℎ−^𝑛+1_𝑗₌₋₁𝛿_𝑗𝛼𝐵_𝑗,𝜙_𝑖_ℎ=0,𝑖=0,1,2,…,𝑛,_𝑛+1𝑗₌₋₁𝛾_𝑗𝑏𝐵_𝑗,𝜙_𝑖_ℎ+^𝑛+1_𝑗₌₋₁𝛿_𝑗𝐵^𝑗,𝜙_𝑖_ℎ=⟨𝑓,𝜙_𝑖⟩_ℎ,𝑖=0,1,2,…,𝑛,(2.15) with the corresponding equations: ^𝑛⁺¹_𝑗=−1𝛾_𝑗𝐵_𝑗(0)=0,^𝑛+1_𝑗₌₋₁𝛾_𝑗𝐵_𝑗(1)=0,_𝑛+1𝑗₌₋₁𝛿_𝑗𝐵_𝑗(0)=0,^𝑛+1_𝑗₌₋₁𝛿_𝑗𝐵_𝑗(0)=0,(2.16) referring to the zero-boundary conditions: 𝑢_ℎ(0)=0,𝑢_ℎ(1)=0,𝑣_ℎ(0)=0,𝑣_ℎ(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,𝑏∈𝐶⁴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(𝑥). It can be proved that the quadrature rule in (2.3) has an error bound of the form 𝐸_ℎ(𝑔)=||||𝒬_ℎ(𝑔)−𝑔||||≤𝐶^𝑛_𝑖=1ℎ^4𝑘‖‖𝑔⁽⁴⁾‖‖_{𝐿1(𝐼𝑘)}.(4.2) This follows from Peano’s kernel theorem (see [11]).

The following inequality is frequently used in our analysis. If 𝑣∈𝑊^𝑚𝑝(𝐸) with 𝑝∈[1,∞], then there exists a positive constant 𝐶 depending only on 𝑚 such that, for any 𝛿satisfying 0<𝛿≤|𝐸|≤1,‖𝑣‖_𝑊^𝑖𝑝_(𝐸)≤𝐶𝛿^𝑚−𝑖‖𝑣‖_𝑊^𝑚𝑝_(𝐸)+𝛿^−𝑖‖𝑣‖_{𝐿𝑝(𝐸)},0≤𝑖≤𝑚−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: 𝐿𝑣∶=𝑣^.(4.4) The adjoint operator 𝐿^∗ with corresponding adjoint boundary condition is defined as follows: 𝜙(0𝐿)^∗=𝜙𝜙=(1𝜙)^=,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 ‖𝐿^∗𝑢‖_{𝐿𝑞(𝐼)}≥𝐶‖𝑢‖_𝑊^2𝑞_(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], ‖𝜙‖_{𝐿∞(𝐼𝑘)}≤‖𝜙‖_𝑊^1𝑝_(𝐼𝑘);1≤𝑝≤∞,𝜙∈𝑊^1𝑝𝐼_𝑘.(4.7)

5. Convergence Analysis

Hereafter throughout this section, for 𝑝 and 𝑞 with 1≤𝑝,𝑞≤∞,s  and 𝑝⁻¹+𝑞⁻¹=1, we use the following notations: ‖𝑣‖_0,𝑝=‖𝑣‖_𝐿𝑝,‖𝑣‖_𝑠,𝑝=‖𝑣‖_𝑊^𝑠𝑝,‖𝑣‖_{𝑠,𝑝,𝑘}=‖𝑣‖_𝑊^𝑠𝑝_(𝐼𝑘).(5.1) Let us denote the error between 𝑢 and 𝑢_ℎ by 𝜀_ℎ and the error between 𝑣 and 𝑣_ℎ by 𝑒_ℎ, respectively, that is, 𝜀_ℎ=𝑢−𝑢_ℎ and 𝑒_ℎ=𝑣−𝑣_ℎ. Using (2.11) and (1.5), we obtain the following error equations: 𝑒^ℎ,𝜙_ℎ_ℎ=𝑣^−𝑣^ℎ,𝜙_ℎ_ℎ=𝑣^,𝜙_ℎ_ℎ−⟨𝑓−𝑏𝑢_ℎ,𝜙_ℎ⟩_ℎ=−𝑏𝑢−𝑢_ℎ,𝜙_ℎ_ℎ=−⟨𝑏𝜀_ℎ,𝜙_ℎ⟩_ℎ,(5.2) and therefore we get 𝑒^ℎ,𝜙_ℎ_ℎ=−⟨𝑏𝜀_ℎ,𝜙_ℎ⟩_ℎ,𝜙_ℎ∈𝑆_ℎ,1.(5.3) Further, using (2.10) and (1.4),𝜀^ℎ,𝜙_ℎ_ℎ=𝑢^−𝑢^ℎ,𝜙_ℎ_ℎ=𝛼𝑣−𝑣_ℎ,𝜙_ℎ_ℎ=⟨𝛼𝑒_ℎ,𝜙_ℎ⟩_ℎ,(5.4) and therefore we have 𝜀^ℎ,𝜙_ℎ_ℎ=⟨𝛼𝑒_ℎ,𝜙_ℎ⟩_ℎ,𝜙_ℎ∈𝑆_ℎ,1.(5.5) The following lemma gives estimates for the error in the quadrature rule for the term (𝑒^ℎ𝜒_ℎ) and (𝜀^ℎ𝜒_ℎ) for 𝜒_ℎ∈𝑆_ℎ,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 𝜒_ℎ∈𝑆_ℎ,1 and h sufficiently small, (a)𝐸_ℎ(𝑒^ℎ𝜒_ℎ)≤𝐶ℎ⁴‖𝑣‖_6,𝑝‖𝜒_ℎ‖_1,𝑞, (b)𝐸_ℎ(𝑒^ℎ𝜒_ℎ)≤𝐶ℎ³‖𝑣‖_6,𝑝‖𝜒_ℎ‖_0,𝑞, (c)𝐸_ℎ(𝜀^ℎ𝜒_ℎ)≤𝐶ℎ⁴‖𝑢‖_6,𝑝‖𝜒_ℎ‖_1,𝑞, (d)𝐸_ℎ(𝜀^ℎ𝜒_ℎ)≤𝐶ℎ³‖𝑢‖_6,𝑝‖𝜒_ℎ‖_0,𝑞.

The following result gives estimate for 𝜀_ℎ(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_ℎ be the corresponding discrete Petrov-Galerkin solution defined through (2.10). Then, the error 𝜀_ℎ=𝑢−𝑢_ℎ satisfies ||𝜀_ℎ𝑥||≤𝐶ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_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 𝐿_𝑝(𝐼)⋂𝐶(𝐼) satisfying the following auxiliary problem: Φ(0)=Φ(Φ1)^==00,,𝑥Φ∈_′−𝐼−𝑥,𝑥−Φ^+𝑥=−1.(5.7) The above problem has a solution. For example, Φ(𝑥)=𝑥−1𝑥,0≤𝑥≤𝑥,𝑥(𝑥−1),𝑥≤𝑥≤1(5.8) satisfies the above differential equation, the boundary conditions, and the jump condition.
Let us define Ψ as follows: Ψ(𝑥)=Φ^0,,𝑥at∈𝑥𝐼=−𝑥,𝑥.(5.9) Then, Ψ=0  a.e. on 𝐼. We first multiply 𝜀_ℎ with Ψ and then integrate over 𝐼. On applying integration by parts, using the fact that 𝜀_ℎ(0)=𝜀_ℎ(1)=0 and the jump condition for Φ^′, we obtain 0=𝜀_ℎ,Ψ=^𝑥0𝜀_ℎΨ+¹_𝑥𝜀_ℎΨ=^𝑥0𝜀_ℎΦ^+¹_𝑥𝜀_ℎΦ^=𝜀_ℎΦ^^𝑥0−^𝑥0𝜀^ℎΦ^+𝜀_ℎΦ^¹_𝑥−¹_𝑥𝜀^ℎΦ^=𝜀_ℎ𝑥Φ^−𝑥−Φ^+𝑥−^𝑥0𝜀^ℎΦ^−¹_𝑥𝜀^ℎΦ^=−𝜀_ℎ𝑥−^𝑥0𝜀^ℎΦ^−¹_𝑥𝜀^ℎΦ^.(5.10) Applying integration by parts once again, using boundary condition for Φ and the continuity of Φ, we obtain 0=−𝜀_ℎ𝑥−𝜀^ℎΦ^𝑥0−^𝑥0𝜀^ℎΦ+𝜀^ℎΦ¹_𝑥−¹_𝑥𝜀^ℎΦ=−𝜀_ℎ𝑥+𝜀^ℎ,Φ,(5.11) that is, 𝜀_ℎ(𝑥)=(𝜀^ℎ,Φ). Let Φ_ℎ be the linear interpolant of Φ. Then, we have 𝜀_ℎ𝑥=𝜀^ℎ,Φ−Φ_ℎ+𝜀^ℎ,Φ_ℎ−𝜀^ℎ,Φ_ℎ_ℎ+𝜀^ℎ,Φ_ℎ_ℎ||𝜀_ℎ𝑥||≤||𝜀^ℎ,Φ−Φ_ℎ||+||𝐸_ℎ𝜀^ℎΦ_ℎ||+||𝜀^ℎ,Φ_ℎ_ℎ||≤𝑇₁+𝑇₂+𝑇₃.(5.12) We know that ‖‖Φ_ℎ‖‖_1,𝑞≤‖‖Φ−Φ_ℎ‖‖_1,𝑞+‖Φ‖_1,𝑞≤𝐶ℎ‖Φ‖_2,𝑞+‖Φ‖_2,𝑞≤𝐶‖Φ‖_2,𝑞.(5.13) We now compute the estimates for the terms 𝑇₁, 𝑇₂, and 𝑇₃ as follows: 𝑇₁=||𝜀^ℎ,Φ−Φ_ℎ||≤‖‖𝜀^ℎ‖‖_0,𝑝‖‖Φ−Φ_ℎ‖‖_0,𝑞≤𝐶ℎ²‖‖𝜀_ℎ‖‖_2,𝑝‖Φ‖_2,𝑞.(5.14) Using Lemma 5.1(c) and (5.13), we obtain 𝑇₂=||𝐸_ℎ𝜀^ℎΦ_ℎ||≤𝐶ℎ⁴‖𝑢‖_6,𝑝‖Φ‖_2,𝑞.(5.15) Using (5.5), (2.3), and the Sobolev embedding theorem (4.7) locally on 𝐼_𝑘 for both ‖𝑒_ℎ‖_0,∞,𝑘 and ‖Φ_ℎ‖_0,∞,𝑘, we have 𝑇₃=||𝜀^ℎ,Φ_ℎ_ℎ||=||⟨𝛼𝑒_ℎ,Φ_ℎ⟩_ℎ||≤𝐶^𝑛_𝑘=1ℎ_𝑘2‖‖𝑒^ℎ‖‖_0,∞,𝑘‖‖Φ_ℎ‖‖_0,∞,𝑘≤𝐶^𝑛_𝑘=1ℎ_𝑘2‖‖𝑒^ℎ‖‖_1,𝑝,𝑘‖‖Φ_ℎ‖‖_1,𝑞,𝑘.(5.16) Using Hölder's inequality for sums and (5.13), we have 𝑇₃≤𝐶ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖‖Φ_ℎ‖‖_1,𝑞≤𝐶ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖Φ‖_2,𝑞.(5.17) For Φ satisfying the auxiliary problem, it is easy to verify that ‖Φ‖_2,𝑞≤𝐾, where 𝐾 is a constant not depending on ℎ.
Using 𝑇₁, 𝑇₂, and 𝑇₃ in (5.12), we have ||𝜀_ℎ𝑥||≤𝐶ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝.(5.18) This completes the proof.

In the following lemma, we initially compute the error (𝑣−𝑣_ℎ) in terms of (𝑢−𝑢_ℎ), and then later on we establish an optimal estimate of error (𝑣−𝑣_ℎ) independent of (𝑢−𝑢_ℎ).

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_ℎ and 𝑣_ℎ be the corresponding discrete Petrov-Galerkin solutions defined through (2.10) and (2.11), respectively. Then the estimates of the errors 𝑒_ℎ=𝑣−𝑣_ℎ in 𝐿_𝑝, 𝑊^1𝑝, and 𝑊^2𝑝 norms are given as follows: ‖‖‖‖𝑒𝑒_ℎℎ‖‖_0,𝑝≤𝐶ℎ⁴‖𝑣‖_6,𝑝+ℎ⁵‖𝑢‖_6,𝑝+ℎ³‖‖𝜀_ℎ‖‖_2,𝑝,‖‖𝑒_ℎ‖‖_1,𝑝≤𝐶ℎ³‖𝑣‖_6,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝,‖‖_2,𝑝≤𝐶ℎ²‖𝑣‖_6,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝.(5.19)

Proof. Let 𝜂 be an arbitrary element of 𝐿_𝑞, and let 𝜙∈𝑊^2𝑞 be the solution of the auxiliary problem 𝜙(0)𝐿=^∗𝜙𝜙=(1𝜂),=0.(5.20) We now have 𝑒_ℎ,𝜂==𝑒𝑒_ℎℎ,𝐿^∗𝜙=𝐿𝑒_ℎ,𝜙=𝑒^ℎ,𝜙−𝜙_ℎ+𝑒^ℎ,𝜙_ℎ,𝜙−𝜙_ℎ+𝑒^ℎ,𝜙_ℎ−𝑒^ℎ,𝜙_ℎ_ℎ+𝑒^ℎ,𝜙_ℎ_ℎ||𝑒_ℎ=𝑒^ℎ,𝜙−𝜙_ℎ+𝐸_ℎ𝑒^ℎ𝜙_ℎ+𝑒^ℎ,𝜙_ℎ_ℎ,,𝜂||≤≤𝑇||₄𝑒^ℎ,𝜙−𝜙_ℎ||+||𝐸_ℎ𝑒^ℎ𝜙_ℎ||+||𝑒^ℎ,𝜙_ℎ_ℎ||+𝑇₅+𝑇₆,(5.21) where 𝜙_ℎ∈𝑆_ℎ,1 is the linear interpolant of 𝜙.
We know that ‖‖𝜙_ℎ‖‖_1,𝑞≤‖‖𝜙−𝜙_ℎ‖‖_1,𝑞+‖𝜙‖_1,𝑞≤𝐶ℎ‖𝜙‖_2,𝑞+‖𝜙‖_2,𝑞≤𝐶‖𝜙‖_2,𝑞.(5.22) We shall compute the estimates for the terms 𝑇₄, 𝑇₅, and 𝑇₆ as follows: 𝑇₅𝑇₄=||𝑒^ℎ,𝜙−𝜙_ℎ||≤‖‖𝑒^ℎ‖‖_0,𝑝‖‖𝜙−𝜙_ℎ‖‖_0,𝑞≤𝐶ℎ²‖‖𝑒_ℎ‖‖_2,𝑝‖𝜙‖_2,𝑞,=||𝐸_ℎ𝑒^ℎ𝜙_ℎ||≤𝐶ℎ⁴‖𝑣‖_6,𝑝‖‖𝜙_ℎ‖‖_1,𝑞≤𝐶ℎ⁴‖𝑣‖_6,𝑝‖𝜙‖_2,𝑞byLemma5.1(a),(5.22).(5.23) Using (5.3), (2.3), and the Sobolev embedding theorem (4.7) locally on 𝐼_𝑘 for ‖𝜙_ℎ‖_0,∞,𝑘, we have 𝑇₆=||𝑒^ℎ,𝜙_ℎ_ℎ||=||−⟨𝑏𝜀_ℎ,𝜙_ℎ⟩_ℎ||≤𝐶^𝑛_𝑘=1ℎ_𝑘2‖‖𝜀^ℎ‖‖_0,∞,𝑘‖‖𝜙_ℎ‖‖_0,∞,𝑘≤𝐶^𝑛_𝑘=1ℎ_𝑘2‖‖𝜀^ℎ‖‖_0,∞,𝑘‖‖𝜙_ℎ‖‖_1,𝑞,𝑘.(5.24) Using Hölder's inequality for sums, Lemma 5.2, and (5.22), we obtain 𝑇₆≤𝐶ℎℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖‖𝜙_ℎ‖‖_1,𝑞≤𝐶ℎ³‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁵‖𝑢‖_6,𝑝+ℎ²‖‖𝑒_ℎ‖‖_1,𝑝‖𝜙‖_2,𝑞.(5.25) Substituting 𝑇₄, 𝑇₅, and 𝑇₆ in (5.21), we have ||𝑒_ℎ,𝜂||≤𝐶ℎ²‖‖𝑒_ℎ‖‖_2,𝑝+ℎ⁴‖𝑣‖_6,𝑝+ℎ³‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁵‖𝑢‖_6,𝑝+ℎ²‖‖𝑒_ℎ‖‖_1,𝑝‖𝜙‖_2,𝑞.(5.26) Using (4.6) and the regularity of the auxiliary problem, we have ‖𝜙‖_2,𝑞≤𝐶‖𝜂‖_0,𝑞. Since 𝜂∈𝐿_𝑞 is arbitrary, we have ‖‖𝑒_ℎ‖‖_0,𝑝≤𝐶ℎ²‖‖𝑒_ℎ‖‖_2,𝑝+ℎ³‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑣‖_6,𝑝+ℎ⁵‖𝑢‖_6,𝑝.(5.27) We now estimate ‖𝑒^ℎ‖ via a projection argument. Let 𝑃_ℎ be the orthogonal projection onto 𝑆_ℎ,1 with respect to 𝐿₂ inner product defined by 𝑣^−𝑃_ℎ𝑣^,𝜓_ℎ=0,𝜓_ℎ∈𝑆_ℎ,1.(5.28) The domain of 𝑃_ℎ 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, ‖‖𝑃_ℎ𝑣‖‖_0,𝑝≤𝐶‖𝑣‖_0,𝑝.(5.29) Then the error 𝑒^ℎ can be interpreted in terms of the error of the above projection: ‖‖𝑒^ℎ‖‖_0,𝑝=‖‖𝑣^−𝑣^ℎ‖‖_0,𝑝≤‖‖𝑣^−𝑃_ℎ𝑣^‖‖_0,𝑝+‖‖𝑃_ℎ𝑣^−𝑣^ℎ‖‖_0,𝑝.(5.30) From the stability property (5.29), the error in the projection follows as in de Boor [14], that is, ‖‖𝑣^−𝑃_ℎ𝑣^‖‖_0,𝑝≤𝐶ℎ²‖‖𝑣^‖‖_2,𝑝≤𝐶ℎ²‖𝑣‖_4,𝑝.(5.31) Then the remaining task is to compute the estimate of ‖𝑃_ℎ𝑣^−𝑣^ℎ‖_0,𝑝.
For 𝜓_ℎ∈𝑆_ℎ,1, 𝑃_ℎ𝑣^−𝑣^ℎ,𝜓_ℎ==𝑃𝑃_ℎℎ𝑣^−𝑣^+𝑣^−𝑣^ℎ,𝜓_ℎ=𝑣_𝑣^−𝑣^,𝜓_ℎ+𝑣^−𝑣^ℎ,𝜓_ℎ𝑃−𝑣^ℎ,𝜓_ℎusing(5.28),_ℎ𝑣^−𝑣^ℎ,𝜓_ℎ=𝑒^ℎ,𝜓_ℎ=𝑒^ℎ,𝜓_ℎ−𝑒^ℎ,𝜓_ℎ_ℎ+𝑒^ℎ,𝜓_ℎ_ℎ||𝑃_ℎ=𝐸_ℎ𝑒^ℎ𝜓_ℎ+𝑒^ℎ,𝜓_ℎ_ℎ,𝑣^−𝑣^ℎ,𝜓_ℎ||≤||𝐸_ℎ𝑒^ℎ𝜓_ℎ||+||𝑒^ℎ,𝜓_ℎ_ℎ||≤𝑇₇+𝑇₈.(5.32) We shall compute the estimates for the terms 𝑇₇ and 𝑇₈𝑇₇=||𝐸_ℎ𝑒^ℎ𝜓_ℎ||≤𝐶ℎ³‖𝑣‖_6,𝑝‖‖𝜓_ℎ‖‖_0,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^ℎ,𝜓_ℎ_ℎ||≤𝐶ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖‖𝜓_ℎ‖‖_0,𝑞,(5.34) where we have used the inverse inequality ‖𝜓_ℎ‖_1,𝑞,𝑘≤ℎ^𝑘−1‖𝜓_ℎ‖_0,𝑞,𝑘 locally. Using 𝑇₇ and 𝑇₈ in (5.32), we get ||𝑃_ℎ𝑣^−𝑣^ℎ,𝜓_ℎ||≤𝐶ℎ³‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖‖𝜓_ℎ‖‖_0,𝑞.(5.35) We now show the above inequality for 𝜂∈𝐿_𝑞 to obtain ‖𝑃_ℎ𝑣^−𝑣^ℎ‖_0,𝑝.
Now let 𝜂 be an arbitrary element of 𝐿_𝑞. Then since 𝑣^ℎ∈𝑆_ℎ,1, it follows from the definition of 𝑃_ℎ𝜂, (5.35), and (5.29) with 𝑝 replaced by 𝑞, that ||𝑃_ℎ0=𝑃_ℎ𝑣^−𝑣^ℎ,𝜂−𝑃_ℎ𝜂,𝑣^−𝑣^ℎ,𝜂||=||𝑃_ℎ𝑣^−𝑣^ℎ,𝑃_ℎ𝜂||≤𝐶ℎ³‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖‖𝑃_ℎ𝜂‖‖_0,𝑞‖‖𝑃_ℎ≤𝐶ℎ³‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖𝜂‖_0,𝑞,𝑣^−𝑣^ℎ‖‖_0,𝑝≤𝐶ℎ³‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝.(5.36) Now, from (5.30), (5.31), and (5.36), we conclude that ‖‖𝑒^ℎ‖‖_0,𝑝≤≤𝐶𝐶ℎ²ℎ‖₂𝑣‖_4,𝑝+𝐶ℎ³‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝.(5.37) Now, using the fact ‖𝑒_ℎ‖_2,𝑝≤‖𝑒_ℎ‖_1,𝑝+‖𝑒^ℎ‖_0,𝑝 and the above estimate, we have ‖‖𝑒_ℎ‖‖_2,𝑝≤≤𝐶‖‖𝑒_ℎ‖‖‖‖𝑒_1,𝑝ℎ+𝐶ℎ²‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝≤𝐶‖‖𝑒_ℎ‖‖_1,𝑝+ℎ²‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝‖‖_1,𝑝+ℎ²‖𝑣‖_6,𝑝+ℎ²‖𝜀_ℎ‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝.(5.38) Now using (4.3) with 𝑚=2 and 𝑖=1, we have ‖‖𝑒_ℎ‖‖_1,𝑝≤𝐶ℎ⁻¹‖‖𝑒_ℎ‖‖_0,𝑝+ℎ‖‖𝑒_ℎ‖‖_2,𝑝.(5.39) Substituting (5.39) in the above expression, we obtain ‖‖𝑒_ℎ‖‖_2,𝑝≤𝐶ℎ⁻¹‖‖𝑒_ℎ‖‖_0,𝑝+ℎ‖‖𝑒_ℎ‖‖_2,𝑝+ℎ²‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝.(5.40) For sufficiently small ℎ, we have ‖‖𝑒_ℎ‖‖_2,𝑝≤𝐶ℎ⁻¹‖‖𝑒_ℎ‖‖_0,𝑝+ℎ²‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝.(5.41) Using (5.41) in (5.27), ‖‖𝑒_ℎ‖‖_0,𝑝≤𝐶ℎ²ℎ⁻¹‖‖𝑒_ℎ‖‖_0,𝑝+ℎ²‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ⁴‖𝑣‖_6,𝑝+ℎ⁵‖𝑢‖_6,𝑝+ℎ³‖‖𝜀_ℎ‖‖_2,𝑝.(5.42) For sufficiently small ℎ, we get ‖‖𝑒_ℎ‖‖_0,𝑝≤𝐶ℎ⁴‖𝑣‖_6,𝑝+ℎ⁵‖𝑢‖_6,𝑝+ℎ³‖‖𝜀_ℎ‖‖_2,𝑝.(5.43) Using (5.43) in (5.41), we have ‖‖𝑒_ℎ‖‖_2,𝑝≤≤𝐶𝐶ℎℎ⁻¹₂ℎ⁴‖𝑣‖_6,𝑝+ℎ⁵‖𝑢‖_6,𝑝+ℎ³‖‖𝜀_ℎ‖‖_2,𝑝+ℎ²‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝.(5.44) Using (5.43) and (5.44) in (5.39), we have ‖‖𝑒_ℎ‖‖_1,𝑝≤≤𝐶𝐶ℎℎ⁻¹₃ℎ⁴‖𝑣‖_6,𝑝+ℎ⁵‖𝑢‖_6,𝑝+ℎ³‖‖𝜀_ℎ‖‖_2,𝑝+ℎℎ²‖𝑣‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝‖𝑣‖_6,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ²‖‖𝜀_ℎ‖‖_2,𝑝.(5.45) Equations (5.43), (5.44), and (5.45) give the required result.

We now compute the error estimate of 𝜀_ℎ in 𝐿_𝑝,𝑊^1𝑝,and 𝑊^2𝑝 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_ℎ and 𝑣_ℎ be the corresponding discrete Petrov-Galerkin solutions defined through (2.10) and (2.11), respectively. Then the estimates of the errors 𝜀_ℎ=𝑢−𝑢_ℎ in 𝐿_𝑝,𝑊^1𝑝and 𝑊^2𝑝 norms are given as follows: ‖‖‖‖𝜀𝜀_ℎℎ‖‖_0,𝑝≤𝐶ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝,‖‖𝜀_ℎ‖‖_1,𝑝≤𝐶ℎ³‖𝑢‖_6,𝑝+‖‖𝑒_ℎ‖‖_1,𝑝,‖‖_2,𝑝≤𝐶ℎ²‖𝑢‖_6,𝑝+‖‖𝑒_ℎ‖‖_1,𝑝.(5.46)

Proof. Let 𝜌 be an arbitrary element of 𝐿_𝑞, and let 𝜙∈𝑊^2𝑞 be the unique solution of the auxiliary problem 𝜙(0)𝐿=^∗𝜙𝜙=(1𝜌),=0.(5.47) Then we have 𝜀_ℎ,𝜌=𝜀_ℎ,𝐿^∗𝜙=𝐿𝜀_ℎ,𝜙=𝜀^ℎ,𝜙=𝜀^ℎ,𝜙−𝜙_ℎ+𝜀^ℎ,𝜙_ℎ−𝜀^ℎ,𝜙_ℎ_ℎ+𝜀^ℎ,𝜙_ℎ_ℎ,(5.48) where 𝜙_ℎ∈𝑆_ℎ,1 is a linear interpolant of 𝜙, ||𝜀_ℎ,𝜌||≤||𝜀^ℎ,𝜙−𝜙_ℎ||+||𝐸_ℎ𝜀^ℎ𝜙_ℎ||+||𝜀^ℎ,𝜙_ℎ_ℎ||≤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: 𝑇𝑇₉₁₀≤𝐶ℎ²‖‖𝜀_ℎ‖‖_2,𝑝‖𝜙‖_2,𝑞,≤𝐶ℎ⁴‖𝑢‖_6,𝑝‖𝜙‖_2,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 𝐼_𝑘 for ‖𝜙_ℎ‖_0,∞,𝑘 and ‖𝑒_ℎ‖_0,∞,𝑘 to estimate 𝑇₁₁, we have 𝑇₁₁=||𝜀^ℎ,𝜙_ℎ_ℎ||=||⟨𝛼𝑒_ℎ,𝜙_ℎ⟩_ℎ||≤𝐶^𝑛_𝑘=1ℎ_𝑘2‖‖𝑒^ℎ‖‖_0,∞,𝑘‖‖𝜙_ℎ‖‖_0,∞,𝑘≤𝐶^𝑛_𝑘=1ℎ_𝑘2‖‖𝑒^ℎ‖‖_0,∞,𝑘‖‖𝜙_ℎ‖‖_1,𝑞,𝑘≤𝐶^𝑛_𝑘=1ℎ_𝑘2‖‖𝑒^ℎ‖‖_1,𝑝,𝑘‖‖𝜙_ℎ‖‖_1,𝑞,𝑘.(5.51) Further, using Hölder's inequality for sums and (5.22), we obtain 𝑇₁₁≤𝐶ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖‖𝜙_ℎ‖‖_1,𝑞≤𝐶ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖𝜙‖_2,𝑞.(5.52) Substituting the estimates 𝑇₉, 𝑇₁₀, and 𝑇₁₁ in (5.49), we obtain ||𝜀_ℎ,𝜌||≤𝐶ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝‖𝜙‖_2,𝑞.(5.53) Using (4.6) and regularity of the auxiliary problem, we have ‖𝜙‖_2,𝑞≤𝐶‖𝜌‖_𝑜,𝑞. Since 𝜌∈𝐿_𝑞 is arbitrary, we have ‖‖𝜀_ℎ‖‖_0,𝑝≤𝐶ℎ²‖‖𝜀_ℎ‖‖_2,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝.(5.54) The estimate of ‖𝜀^ℎ‖_0,𝑝 can be obtained through a projection argument as mentioned in Lemma 5.3 as ‖‖𝜀^ℎ‖‖_0,𝑝≤𝐶ℎ²‖𝑢‖_6,𝑝+‖‖𝑒_ℎ‖‖_1,𝑝,(5.55) where we have used Lemma 5.1(d). In a similar manner we can compute the estimates for ‖𝜀_ℎ‖_0,𝑝, ‖𝜀_ℎ‖_1,𝑝 and ‖𝜀_ℎ‖_2,𝑝 as ‖‖‖‖𝜀𝜀_ℎℎ‖‖_0,𝑝≤𝐶ℎ⁴‖𝑢‖_6,𝑝+ℎ‖‖𝑒_ℎ‖‖_1,𝑝,‖‖𝜀_ℎ‖‖_1,𝑝≤𝐶ℎ³‖𝑢‖_6,𝑝+‖‖𝑒_ℎ‖‖_1,𝑝,‖‖_2,𝑝≤𝐶ℎ²‖𝑢‖_6,𝑝+‖‖𝑒_ℎ‖‖_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 𝑢 and 𝑣 satisfy (1.4) and (1.5), respectively, with (4.1). Assume also that 𝑢∈𝑊^6𝑝 and 𝑣∈𝑊^6𝑝, where 𝑝∈[1,∞]. Then (2.10) and (2.11) have unique solutions 𝑢_ℎ∈𝑆⁰_ℎ,3 and 𝑣_ℎ∈𝑆⁰_ℎ,3, respectively, and for ℎ sufficiently small, one has ‖‖𝑣−𝑣‖‖_ℎ𝑢−𝑢_ℎ‖‖_𝑖,𝑝≤𝐶ℎ^4−𝑖‖𝑢‖_6,𝑝+‖𝑣‖_6,𝑝,‖‖_𝑖,𝑝≤𝐶ℎ^4−𝑖‖𝑢‖_6,𝑝+‖𝑣‖_6,𝑝,𝑖=0,1,2.(5.57)

Proof. Assume temporarily that solutions 𝑢_ℎ and 𝑣_ℎ of (2.10) and (2.11), respectively, exist. Using (5.46) in (5.45), we obtain ‖‖𝑒_ℎ‖‖_1,𝑝≤𝐶ℎ³‖𝑣‖_6,𝑝+ℎ⁴‖𝑢‖_6,𝑝+ℎ²ℎ²‖𝑢‖_6,𝑝+‖𝑒_ℎ‖_1,𝑝.(5.58) For sufficiently small ℎ, we have ‖‖𝑒_ℎ‖‖_1,𝑝≤𝐶ℎ³‖𝑣‖_6,𝑝+ℎ⁴‖𝑢‖_6,𝑝.(5.59) An application of the above in (5.46), we get ‖‖𝜀_ℎ‖‖_2,𝑝≤𝐶ℎ²‖𝑢‖_6,𝑝+ℎ³‖𝑣‖_6,𝑝.(5.60) Apply (5.59) in (5.56) to have ‖‖𝜀_ℎ‖‖_0,𝑝≤𝐶ℎ⁴‖𝑢‖_6,𝑝+ℎ⁴‖𝑣‖_6,𝑝.(5.61) Use (5.60) in (5.43) to get ‖‖𝑒_ℎ‖‖_0,𝑝≤𝐶ℎ⁴‖𝑣‖_6,𝑝+ℎ⁵‖𝑢‖_6,𝑝.(5.62) Using (5.60) in (5.44), we obtain ‖‖𝑒_ℎ‖‖_2,𝑝≤𝐶ℎ²‖𝑣‖_6,𝑝+ℎ⁴‖𝑢‖_6,𝑝.(5.63) Using (5.61) and (5.60) in (5.39) with 𝑒_ℎ replaced by 𝜀_ℎ, we have ‖‖𝜀_ℎ‖‖_1,𝑝≤𝐶ℎ³‖𝑢‖_6,𝑝+ℎ³‖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 𝑢_ℎ and 𝑣_ℎ 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_ℎ and 𝑣_ℎ is square, existence of 𝑢_ℎ∈𝑆⁰_ℎ,3 and 𝑣_ℎ∈𝑆⁰_ℎ,3 for any 𝑓∈𝐶⁰(𝐼) will follow from uniqueness, that is, from the property that the corresponding homogeneous equations have only trivial solutions.

Suppose that 𝑢_ℎ and 𝑣_ℎ corresponding to 𝑢 and 𝑣 satisfy 𝑣𝑢^ℎ−𝛼𝑣_ℎ,𝜒_ℎ=0,_ℎ+𝑏𝑢_ℎ,𝜒_ℎ=0,𝜒_ℎ∈𝑆_ℎ,1.(5.65) It follows from (5.61) and (5.62) (with 𝑢 replaced by 0 and eventually 𝑣≡0) that, for sufficiently small ℎ, ‖‖𝑢_ℎ‖‖_0,𝑝≤0,‖‖𝑣_ℎ‖‖_0,𝑝≤0,(5.66) and hence 𝑢_ℎ≡0 and 𝑣_ℎ≡0. Thus, uniqueness is proved, and hence existence follows from uniqueness.