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

Authors: L. Jones Tarcius Doss; A. P. Nandini
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## 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 ordera priori error estimates are obtained without any restriction on the mesh.


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## Body

## 1. Introduction

In this paper, we develop a quadrature-based Petrov-Galerkin mixed finite element method for the following fourth-order boundary value problem:(1.1)d2dx2[a(x)d2udx2]+b(x)u=f(x),x∈I=(0,1),
subject to the boundary conditions (1.2)u(0)=0,u(1)=0;u″(0)=0,u″(1)=0,
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(1.3)u″=αv,x∈I.
Then the differential equation (1.1) with the boundary conditions (1.2) can be written as a coupled system of equations as follows: (1.4)u″=αv,x∈I,withu(0)=u(1)=0,(1.5)v″+bu=f,x∈I,withv(0)=v(1)=0.
In this paper, the error analysis will take place in the usual Sobolev space Wpm(I) defined on the domain I=(0,1) with Hm(I) denoting W2m(I). The Sobolev norms are given below. For an open interval E and a non negative integer m, (1.6)‖v‖Wpm(E)=(∑i=0m‖v(i)‖Lp(E)p)1/p,if1≤p<∞,=max⁡1≤i≤n‖v(i)‖L∞(E),ifp=∞.
We suppress the dependence of the norms on I when E=I. Further, H0m(I) denotes the function space (1.7){ϕ∈Hm(I):ϕ(0)=ϕ(1)=0}.


## 2. Continuous and DiscreteH1-Galerkin Formulation

Givenn>1, let (2.1)Πn:0=x0<x1<⋯<xn=1
be an arbitrary partition of [0,1] with the property that h→0 as n→∞, where h=max1≤k≤nhk and hk=xk-xk-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: (2.2)(u,v)=∫uvdx,〈u,v〉h=Qh(uv),
where 𝒬h is the fourth-order Gaussian quadrature rule: (2.3)Qh(g):=12∑i=1nhk[g(xk,1)+g(xk,2)].
Here, xk,i=xk-1+ξihk,i=1,2, are the two Gaussian points in the subinterval [xk-1,xk] with ξ1=(1/2)(1-1/3), ξ2=1-ξ1.Let us now consider the following cubic spline space as trial space:(2.4)Sh,3={φ∈C2(I):φ|Ik∈P3(Ik),k=1,2,…,n},
where Pr(Ik) is the space of polynomials of degree r defined over the kth subinterval Ik=[xk-1,xk].The corresponding space with zero Dirichlet boundary condition is denoted by(2.5)S0h,3={φ∈Sh,3:φ(0)=φ(1)=0}.
Further, let us consider the following piecewise linear space (2.6)Sh,1={φ∈C(I):φ|Ik∈P1(Ik),k=1,2,…,n}
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}∈H02(I) such that (2.7)(u″,ϕ)=(αv,ϕ),ϕ∈H2(0,1),(2.8)(v″+bu,ϕ)=(f,ϕ),ϕ∈H2(0,1).


### 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{uh,vh}∈S0h,3 such that (2.9)(uh″,ϕh)=(αvh,ϕh),ϕh∈Sh,1,(vh″+buh,ϕh)=(f,ϕh),ϕh∈Sh,1.
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{uh,vh}∈S0h,3 such that (2.10)〈uh″,ϕh〉h=〈αvh,ϕh〉h,ϕh∈Sh,1,(2.11)〈vh″+buh,ϕh〉h=〈f,ϕh〉h,ϕh∈Sh,1.
The approximate solutions uh and vh without any conditions on boundary points are expressed as a linear combination of the B-splines as follows: (2.12)uh(x)=∑j=-1n+1γjBj(x),vh(x)=∑j=-1n+1δjBj(x),
where the jth basis Bj(x) of the cubic B-splines space Sh,3 for j=-1,0,1,2,…,n,n+1 is given below: (2.13)Bj(x)={0,ifx≤xj-2,16h3(x-xj-2)3,ifxj-2≤x≤xj-1,16h3(h3+3h2(x-xj-1)+3h(x-xj-1)2-3(x-xj-1)3),ifxj-1≤x≤xj,16h3(h3+3h2(xj+1-x)+3h(xj+1-x)2-3(xj+1-x)3),ifxj≤x≤xj+1,16h3(xj+2-x)3,ifxj+1≤x≤xj+2,0,ifx≥xj+2.Forj=-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 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 Sh,1 for i=0,1,2,…,n is given below: (2.14)ϕi(x)={0,ifx≤xi-1,1h(x-xi-1),ifxi-1≤x≤xi,1h(xi+1-x),ifxi≤x≤xi+1,0,ifx≥xi+1.In a similar manner, fori=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 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: (2.15)∑j=-1n+1γj〈Bj″,ϕi〉h-∑j=-1n+1δj〈αBj,ϕi〉h=0,i=0,1,2,…,n,∑j=-1n+1γj〈bBj,ϕi〉h+∑j=-1n+1δj〈Bj″,ϕi〉h=〈f,ϕi〉h,i=0,1,2,…,n,
with the corresponding equations: (2.16)∑j=-1n+1γjBj(0)=0,∑j=-1n+1γjBj(1)=0,∑j=-1n+1δjBj(0)=0,∑j=-1n+1δjBj(0)=0,
referring to the zero-boundary conditions: (2.17)uh(0)=0,uh(1)=0,vh(0)=0,vh(1)=0.
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.


## 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}∈H02(I) such that (2.7)(u″,ϕ)=(αv,ϕ),ϕ∈H2(0,1),(2.8)(v″+bu,ϕ)=(f,ϕ),ϕ∈H2(0,1).


## 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{uh,vh}∈S0h,3 such that (2.9)(uh″,ϕh)=(αvh,ϕh),ϕh∈Sh,1,(vh″+buh,ϕh)=(f,ϕh),ϕh∈Sh,1.
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{uh,vh}∈S0h,3 such that (2.10)〈uh″,ϕh〉h=〈αvh,ϕh〉h,ϕh∈Sh,1,(2.11)〈vh″+buh,ϕh〉h=〈f,ϕh〉h,ϕh∈Sh,1.
The approximate solutions uh and vh without any conditions on boundary points are expressed as a linear combination of the B-splines as follows: (2.12)uh(x)=∑j=-1n+1γjBj(x),vh(x)=∑j=-1n+1δjBj(x),
where the jth basis Bj(x) of the cubic B-splines space Sh,3 for j=-1,0,1,2,…,n,n+1 is given below: (2.13)Bj(x)={0,ifx≤xj-2,16h3(x-xj-2)3,ifxj-2≤x≤xj-1,16h3(h3+3h2(x-xj-1)+3h(x-xj-1)2-3(x-xj-1)3),ifxj-1≤x≤xj,16h3(h3+3h2(xj+1-x)+3h(xj+1-x)2-3(xj+1-x)3),ifxj≤x≤xj+1,16h3(xj+2-x)3,ifxj+1≤x≤xj+2,0,ifx≥xj+2.Forj=-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 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 Sh,1 for i=0,1,2,…,n is given below: (2.14)ϕi(x)={0,ifx≤xi-1,1h(x-xi-1),ifxi-1≤x≤xi,1h(xi+1-x),ifxi≤x≤xi+1,0,ifx≥xi+1.In a similar manner, fori=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 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: (2.15)∑j=-1n+1γj〈Bj″,ϕi〉h-∑j=-1n+1δj〈αBj,ϕi〉h=0,i=0,1,2,…,n,∑j=-1n+1γj〈bBj,ϕi〉h+∑j=-1n+1δj〈Bj″,ϕi〉h=〈f,ϕi〉h,i=0,1,2,…,n,
with the corresponding equations: (2.16)∑j=-1n+1γjBj(0)=0,∑j=-1n+1γjBj(1)=0,∑j=-1n+1δjBj(0)=0,∑j=-1n+1δjBj(0)=0,
referring to the zero-boundary conditions: (2.17)uh(0)=0,uh(1)=0,vh(0)=0,vh(1)=0.
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 Sections1 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 thatα and b are such that (4.1)α,b∈C4(I¯),
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 (4.2)Eh(g)=|Qh(g)-∫g|≤C∑i=1nhk4‖g(4)‖L1(Ik).
This follows from Peano’s kernel theorem (see [11]).The following inequality is frequently used in our analysis. Ifv∈Wpm(E) with p∈[1,∞], then there exists a positive constant C depending only on m such that, for any δsatisfying 0<δ≤|E|≤1,(4.3)‖v‖Wpi(E)≤C[δm-i‖v‖Wpm(E)+δ-i‖v‖Lp(E)],0≤i≤m-1,
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: (4.4)Lv:=v″.
The adjoint operator L* with corresponding adjoint boundary condition is defined as follows: (4.5)L*ϕ=ϕ″,ϕ(0)=ϕ(1)=0.
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 (4.6)‖L*u‖Lq(I)≥C‖u‖Wq2(I).
We have the following inequality due to the Sobolev embedding theorem; the proof of which can be found in page 97, Adams [12], (4.7)‖ϕ‖L∞(Ik)≤∥ϕ∥Wp1(Ik);1≤p≤∞,ϕ∈Wp1(Ik).


## 5. Convergence Analysis

Hereafter throughout this section, forp and q with 1≤p,q≤∞,s  and p-1+q-1=1, we use the following notations: (5.1)‖v‖0,p=‖v‖Lp,‖v‖s,p=‖v‖Wps,‖v‖s,p,k=‖v‖Wps(Ik).
Let us denote the error between u and uh by εh and the error between v and vh by eh, respectively, that is, εh=u-uh and eh=v-vh. Using (2.11) and (1.5), we obtain the following error equations: (5.2)〈eh″,ϕh〉h=〈v″-vh″,ϕh〉h=〈v″,ϕh〉h-〈f-buh,ϕh〉h=-〈b(u-uh),ϕh〉h=-〈bεh,ϕh〉h,
and therefore we get (5.3)〈eh″,ϕh〉h=-〈bεh,ϕh〉h,ϕh∈Sh,1.
Further, using (2.10) and (1.4),(5.4)〈εh″,ϕh〉h=〈u″-uh″,ϕh〉h=〈α(v-vh),ϕh〉h=〈αeh,ϕh〉h,
and therefore we have (5.5)〈εh″,ϕh〉h=〈αeh,ϕh〉h,ϕh∈Sh,1.
The following lemma gives estimates for the error in the quadrature rule for the term (eh″χh) and (εh″χh) for χh∈Sh,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 allχh∈Sh,1 and h sufficiently small, (a)
Eh(eh″χh)≤Ch4∥v∥6,p∥χh∥1,q,(b)
Eh(eh″χh)≤Ch3∥v∥6,p∥χh∥0,q,(c)
Eh(εh″χh)≤Ch4∥u∥6,p∥χh∥1,q,(d)
Eh(εh″χh)≤Ch3∥u∥6,p∥χh∥0,q.The following result gives estimate forεh(x¯), where x¯ is any arbitrary point in I. This estimate is crucial for our error analysis.Lemma 5.2.
Letu be the weak solution of  (1.4) defined through  (2.7). Further, let  uh be the corresponding discrete Petrov-Galerkin solution defined through  (2.10). Then, the error  εh=u-uh satisfies
(5.6)|εh(x¯)|≤C[h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p],
where x¯ is an arbitrary point in [0,1].Proof.
For a givenx¯∈[0,1], let Φ be an element of Lp(I)⋂C(I) satisfying the following auxiliary problem:
(5.7)Φ″=0,x∈I-{x¯},Φ(0)=Φ(1)=0,Φ-′(x¯)-Φ+′(x¯)=-1.
The above problem has a solution. For example,
(5.8)Φ(x)={(x¯-1)x,0≤x≤x¯,x¯(x-1),x¯≤x≤1
satisfies the above differential equation, the boundary conditions, and the jump condition.
Let us defineΨ as follows:
(5.9)Ψ(x)={Φ″,x∈I-{x¯},0,atx=x¯.
Then, Ψ=0  a.e. on I. We first multiply εh with Ψ and then integrate over I. On applying integration by parts, using the fact that εh(0)=εh(1)=0 and the jump condition for Φ′, we obtain
(5.10)0=(εh,Ψ)=∫0x¯εhΨ+∫x¯1εhΨ=∫0x¯εhΦ″+∫x¯1εhΦ″=[εhΦ′]0x¯-∫0x¯εh′Φ′+[εhΦ′]x¯1-∫x¯1εh′Φ′=εh(x¯)[Φ-′(x¯)-Φ+′(x¯)]-∫0x¯εh′Φ′-∫x¯1εh′Φ′=-εh(x¯)-∫0x¯εh′Φ′-∫x¯1εh′Φ′.
Applying integration by parts once again, using boundary condition for Φ and the continuity of Φ, we obtain
(5.11)0=-εh(x¯)-{[εh′Φ]0x¯-∫0x¯εh″Φ+[εh′Φ]x¯1-∫x¯1εh″Φ}=-εh(x¯)+(εh″,Φ),
that is, εh(x¯)=(εh″,Φ). Let Φh be the linear interpolant of Φ. Then, we have
(5.12)εh(x¯)=(εh″,Φ-Φh)+(εh″,Φh)-〈εh″,Φh〉h+〈εh″,Φh〉h|εh(x¯)|≤|(εh″,Φ-Φh)|+|Eh(εh″Φh)|+|〈εh″,Φh〉h|≤T1+T2+T3.
We know that
(5.13)‖Φh‖1,q≤‖Φ-Φh‖1,q+‖Φ‖1,q≤Ch‖Φ‖2,q+‖Φ‖2,q≤C‖Φ‖2,q.
We now compute the estimates for the terms T1, T2, and T3 as follows:
(5.14)T1=|(εh″,Φ-Φh)|≤‖εh″‖0,p‖Φ-Φh‖0,q≤Ch2‖εh‖2,p‖Φ‖2,q.
Using Lemma 5.1(c) and (5.13), we obtain
(5.15)T2=|Eh(εh″Φh)|≤Ch4‖u‖6,p‖Φ‖2,q.
Using (5.5), (2.3), and the Sobolev embedding theorem (4.7) locally on Ik for both ∥eh∥0,∞,k and ∥Φh∥0,∞,k, we have
(5.16)T3=|〈εh″,Φh〉h|=|〈αeh,Φh〉h|≤C∑k=1nhk2‖eh‖0,∞,k‖Φh‖0,∞,k≤C∑k=1nhk2‖eh‖1,p,k‖Φh‖1,q,k.
Using Hölder's inequality for sums and (5.13), we have
(5.17)T3≤Ch‖eh‖1,p‖Φh‖1,q≤Ch‖eh‖1,p‖Φ‖2,q.
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.
UsingT1, T2, and T3 in (5.12), we have
(5.18)|εh(x¯)|≤C[h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p].
This completes the proof.In the following lemma, we initially compute the error(v-vh) in terms of (u-uh), and then later on we establish an optimal estimate of error (v-vh) independent of (u-uh).Lemma 5.3.
Letu 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  uh and  vh be the corresponding discrete Petrov-Galerkin solutions defined through  (2.10) and  (2.11), respectively. Then the estimates of the errors  eh=v-vh in  Lp,  Wp1, and  Wp2 norms are given as follows:
(5.19)‖eh‖0,p≤C[h4‖v‖6,p+h5‖u‖6,p+h3‖εh‖2,p],‖eh‖1,p≤C[h3‖v‖6,p+h4‖u‖6,p+h2‖εh‖2,p],‖eh‖2,p≤C[h2‖v‖6,p+h4‖u‖6,p+h2‖εh‖2,p].Proof.
Letη be an arbitrary element of Lq, and let ϕ∈Wq2 be the solution of the auxiliary problem
(5.20)L*ϕ=η,ϕ(0)=ϕ(1)=0.
We now have
(5.21)(eh,η)=(eh,L*ϕ)=(Leh,ϕ)=(eh″,ϕ-ϕh)+(eh″,ϕh)=(eh″,ϕ-ϕh)+(eh″,ϕh)-〈eh″,ϕh〉h+〈eh″,ϕh〉h=(eh″,ϕ-ϕh)+Eh(eh″ϕh)+〈eh″,ϕh〉h,|(eh,η)|≤|(eh″,ϕ-ϕh)|+|Eh(eh″ϕh)|+|〈eh″,ϕh〉h|≤T4+T5+T6,
where ϕh∈Sh,1 is the linear interpolant of ϕ.
We know that(5.22)‖ϕh‖1,q≤‖ϕ-ϕh‖1,q+‖ϕ‖1,q≤Ch‖ϕ‖2,q+‖ϕ‖2,q≤C‖ϕ‖2,q.
We shall compute the estimates for the terms T4, T5, and T6 as follows:
(5.23)T4=|(eh″,ϕ-ϕh)|≤‖eh″‖0,p‖ϕ-ϕh‖0,q≤Ch2‖eh‖2,p‖ϕ‖2,q,T5=|Eh(eh″ϕh)|≤Ch4‖v‖6,p‖ϕh‖1,q≤Ch4‖v‖6,p‖ϕ‖2,qbyLemma5.1(a),(5.22).
Using (5.3), (2.3), and the Sobolev embedding theorem (4.7) locally on Ik for ∥ϕh∥0,∞,k, we have
(5.24)T6=|〈eh″,ϕh〉h|=|-〈bεh,ϕh〉h|≤C∑k=1nhk2‖εh‖0,∞,k‖ϕh‖0,∞,k≤C∑k=1nhk2‖εh‖0,∞,k‖ϕh‖1,q,k.
Using Hölder's inequality for sums, Lemma 5.2, and (5.22), we obtain
(5.25)T6≤Ch[h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p]‖ϕh‖1,q≤C[h3‖εh‖2,p+h5‖u‖6,p+h2‖eh‖1,p]‖ϕ‖2,q.
Substituting T4, T5, and T6 in (5.21), we have
(5.26)|(eh,η)|≤C[h2‖eh‖2,p+h4‖v‖6,p+h3‖εh‖2,p+h5‖u‖6,p+h2‖eh‖1,p]‖ϕ‖2,q.
Using (4.6) and the regularity of the auxiliary problem, we have ∥ϕ∥2,q≤C∥η∥0,q. Since η∈Lq is arbitrary, we have
(5.27)‖eh‖0,p≤C(h2‖eh‖2,p+h3‖εh‖2,p+h4‖v‖6,p+h5‖u‖6,p).
We now estimate ∥eh″∥ via a projection argument. Let Ph be the orthogonal projection onto Sh,1 with respect to L2 inner product defined by
(5.28)(v″-Phv″,ψh)=0,ψh∈Sh,1.
The domain of Ph 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,
(5.29)‖Phv‖0,p≤C‖v‖0,p.
Then the error eh″ can be interpreted in terms of the error of the above projection:
(5.30)‖eh″‖0,p=‖v″-vh″‖0,p≤‖v″-Phv″‖0,p+‖Phv″-vh″‖0,p.
From the stability property (5.29), the error in the projection follows as in de Boor [14], that is,
(5.31)‖v″-Phv″‖0,p≤Ch2‖v″‖2,p≤Ch2‖v‖4,p.
Then the remaining task is to compute the estimate of ∥Phv″-vh″∥0,p.
Forψh∈Sh,1,
(5.32)(Phv″-vh″,ψh)=(Phv″-v″+v″-vh″,ψh)=(Phv″-v″,ψh)+(v″-vh″,ψh)=(v″-vh″,ψh)using(5.28),(Phv″-vh″,ψh)=(eh″,ψh)=(eh″,ψh)-〈eh″,ψh〉h+〈eh″,ψh〉h=Eh(eh″ψh)+〈eh″,ψh〉h,|(Phv″-vh″,ψh)|≤|Eh(eh″ψh)|+|〈eh″,ψh〉h|≤T7+T8.
We shall compute the estimates for the terms T7 and T8(5.33)T7=|Eh(eh″ψh)|≤Ch3‖v‖6,p‖ψh‖0,q
by Lemma 5.1(b).
Following the steps of computation involved in the termT6, we obtain the estimate of T8 as
(5.34)T8=|〈eh″,ψh〉h|≤C[h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p]‖ψh‖0,q,
where we have used the inverse inequality ∥ψh∥1,q,k≤hk-1∥ψh∥0,q,k locally. Using T7 and T8 in (5.32), we get
(5.35)|(Phv″-vh″,ψh)|≤C[h3‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p]‖ψh‖0,q.
We now show the above inequality for η∈Lq to obtain ∥Phv″-vh″∥0,p.
Now letη be an arbitrary element of Lq. Then since vh″∈Sh,1, it follows from the definition of Phη, (5.35), and (5.29) with p replaced by q, that
(5.36)0=(Phv″-vh″,η-Phη),|(Phv″-vh″,η)|=|(Phv″-vh″,Phη)|≤C[h3‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p]‖Phη‖0,q≤C[h3‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p]‖η‖0,q,‖Phv″-vh″‖0,p≤C[h3‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p].
Now, from (5.30), (5.31), and (5.36), we conclude that
(5.37)‖eh″‖0,p≤Ch2‖v‖4,p+C[h3‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p]≤C[h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p].
Now, using the fact ∥eh∥2,p≤∥eh∥1,p+∥eh″∥0,p and the above estimate, we have
(5.38)‖eh‖2,p≤‖eh‖1,p+C[h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p]≤C[‖eh‖1,p+h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p]≤C[‖eh‖1,p+h2‖v‖6,p+h2∥εh∥2,p+h4‖u‖6,p].
Now using (4.3) with m=2 and i=1, we have
(5.39)‖eh‖1,p≤C(h-1‖eh‖0,p+h‖eh‖2,p).
Substituting (5.39) in the above expression, we obtain
(5.40)‖eh‖2,p≤C[(h-1‖eh‖0,p+h‖eh‖2,p)+h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p].
For sufficiently small h, we have
(5.41)‖eh‖2,p≤C[h-1‖eh‖0,p+h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p].
Using (5.41) in (5.27),
(5.42)‖eh‖0,p≤C[h2(h-1‖eh‖0,p+h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p)+h4‖v‖6,p+h5‖u‖6,p+h3‖εh‖2,p].
For sufficiently small h, we get
(5.43)‖eh‖0,p≤C[h4‖v‖6,p+h5‖u‖6,p+h3‖εh‖2,p].
Using (5.43) in (5.41), we have
(5.44)‖eh‖2,p≤C[h-1(h4‖v‖6,p+h5‖u‖6,p+h3‖εh‖2,p)+h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p]≤C[h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p].
Using (5.43) and (5.44) in (5.39), we have
(5.45)‖eh‖1,p≤C[h-1(h4‖v‖6,p+h5‖u‖6,p+h3‖εh‖2,p)+h(h2‖v‖6,p+h2‖εh‖2,p+h4‖u‖6,p)]≤C[h3‖v‖6,p+h4‖u‖6,p+h2‖εh‖2,p].
Equations (5.43), (5.44), and (5.45) give the required result.We now compute the error estimate ofεh in Lp,Wp1,and Wp2 norms as has been done in the previous case.Lemma 5.4.
Letu 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  uh and  vh be the corresponding discrete Petrov-Galerkin solutions defined through  (2.10) and  (2.11), respectively. Then the estimates of the errors  εh=u-uh in  Lp,Wp1and  Wp2 norms are given as follows:
(5.46)‖εh‖0,p≤C[h4‖u‖6,p+h‖eh‖1,p],‖εh‖1,p≤C[h3‖u‖6,p+‖eh‖1,p],‖εh‖2,p≤C[h2‖u‖6,p+‖eh‖1,p].Proof.
Letρ be an arbitrary element of Lq, and let ϕ∈Wq2 be the unique solution of the auxiliary problem
(5.47)L*ϕ=ρ,ϕ(0)=ϕ(1)=0.
Then we have
(5.48)(εh,ρ)=(εh,L*ϕ)=(Lεh,ϕ)=(εh″,ϕ)=(εh″,ϕ-ϕh)+(εh″,ϕh)-〈εh″,ϕh〉h+〈εh″,ϕh〉h,
where ϕh∈Sh,1 is a linear interpolant of ϕ,
(5.49)|(εh,ρ)|≤|(εh″,ϕ-ϕh)|+|Eh(εh″ϕh)|+|〈εh″,ϕh〉h|≤T9+T10+T11.
Following the steps involved in the computation of T4 and T5, we obtain the estimates of T9 and T10 as follows:
(5.50)T9≤Ch2‖εh‖2,p‖ϕ‖2,q,T10≤Ch4‖u‖6,p‖ϕ‖2,q,
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 ∥ϕh∥0,∞,k and ∥eh∥0,∞,k to estimate T11, we have
(5.51)T11=|〈εh″,ϕh〉h|=|〈αeh,ϕh〉h|≤C∑k=1nhk2‖eh‖0,∞,k‖ϕh‖0,∞,k≤C∑k=1nhk2‖eh‖0,∞,k‖ϕh‖1,q,k≤C∑k=1nhk2‖eh‖1,p,k‖ϕh‖1,q,k.
Further, using Hölder's inequality for sums and (5.22), we obtain
(5.52)T11≤Ch‖eh‖1,p‖ϕh‖1,q≤Ch‖eh‖1,p‖ϕ‖2,q.
Substituting the estimates T9, T10, and T11 in (5.49), we obtain
(5.53)|(εh,ρ)|≤C[h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p]‖ϕ‖2,q.
Using (4.6) and regularity of the auxiliary problem, we have ∥ϕ∥2,q≤C∥ρ∥o,q. Since ρ∈Lq is arbitrary, we have
(5.54)‖εh‖0,p≤C[h2‖εh‖2,p+h4‖u‖6,p+h‖eh‖1,p].
The estimate of ∥εh″∥0,p can be obtained through a projection argument as mentioned in Lemma 5.3 as
(5.55)‖εh″‖0,p≤C[h2‖u‖6,p+‖eh‖1,p],
where we have used Lemma 5.1(d). In a similar manner we can compute the estimates for ∥εh∥0,p, ∥εh∥1,p and ∥εh∥2,p as
(5.56)‖εh‖0,p≤C[h4‖u‖6,p+h‖eh‖1,p],‖εh‖1,p≤C[h3‖u‖6,p+‖eh‖1,p],‖εh‖2,p≤C[h2‖u‖6,p+‖eh‖1,p].
Using all the estimates from Lemmas 5.3 and 5.4, we have the following main error estimates.Theorem 5.5.
Assume thatu and  v satisfy  (1.4) and  (1.5), respectively, with  (4.1). Assume also that  u∈Wp6 and  v∈Wp6, where  p∈[1,∞]. Then  (2.10) and  (2.11) have unique solutions  uh∈S0h,3 and  vh∈S0h,3, respectively, and for  h sufficiently small, one has
(5.57)‖u-uh‖i,p≤Ch4-i[‖u‖6,p+‖v‖6,p],‖v-vh‖i,p≤Ch4-i[‖u‖6,p+‖v‖6,p],i=0,1,2.Proof.
Assume temporarily that solutionsuh and vh of (2.10) and (2.11), respectively, exist. Using (5.46) in (5.45), we obtain
(5.58)‖eh‖1,p≤C[h3‖v‖6,p+h4∥u∥6,p+h2(h2∥u∥6,p+∥eh∥1,p)].
For sufficiently small h, we have
(5.59)‖eh‖1,p≤C(h3‖v‖6,p+h4‖u‖6,p).
An application of the above in (5.46), we get
(5.60)‖εh‖2,p≤C[h2‖u‖6,p+h3‖v‖6,p].
Apply (5.59) in (5.56) to have
(5.61)‖εh‖0,p≤C[h4‖u‖6,p+h4‖v‖6,p].
Use (5.60) in (5.43) to get
(5.62)‖eh‖0,p≤C[h4‖v‖6,p+h5‖u‖6,p].
Using (5.60) in (5.44), we obtain
(5.63)‖eh‖2,p≤C[h2‖v‖6,p+h4‖u‖6,p].
Using (5.61) and (5.60) in (5.39) with eh replaced by εh, we have
(5.64)‖εh‖1,p≤C[h3‖u‖6,p+h3‖v‖6,p].
The required result can be obtained from estimates (5.59) to (5.64).So far we have assumed temporarily that solutionsuh and vh 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 uh and vh is square, existence of uh∈S0h,3 and vh∈S0h,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 thatuh and vh corresponding to u and v satisfy (5.65)〈uh″-αvh,χh〉=0,〈vh″+buh,χh〉=0,χh∈Sh,1.
It follows from (5.61) and (5.62) (with u replaced by 0 and eventually v≡0) that, for sufficiently small h, (5.66)‖uh‖0,p≤0,‖vh‖0,p≤0,
and hence uh≡0 and vh≡0. Thus, uniqueness is proved, and hence existence follows from uniqueness.



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*Source: 962070-2012-02-13.xml*