# FDM for Elliptic Equations with Bitsadze-Samarskii-Dirichlet Conditions

Authors: Allaberen Ashyralyev; Fatma Songul Ozesenli Tetikoglu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. The first and second-orders of accuracy stable difference schemes for the approximate solution of this nonlocal boundary value problem are presented. The stability estimates, coercivity, and almost coercivity inequalities for solution of these schemes are established. The theoretical statements for the solutions of these nonlocal elliptic problems are supported by results of numerical examples.


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## Body

## 1. Introduction

Many problems in fluid mechanics, dynamics, elasticity, and other areas of engineering, physics, and biological systems lead to partial differential equations of elliptic type. The role played by coercive inequalities in the study of local boundary-value problems for elliptic and parabolic differential equations is well known (see, e.g., [1, 2]).In the present paper, we consider the Bitsadze-Samarskii type nonlocal boundary value problem(1.1)-d2u(t)dt2+Au(t)=f(t),(0<t<1),u′(0)=φ,u′(1)=βu′(λ)+ψ,|β|≤1,0≤λ<1
for elliptic differential equations in a Hilbert space H with self-adjoint positive definite operator A. It is known (see, e.g., [3–11]) that various nonlocal boundary value problems for elliptic equations can be reduced to the boundary value problem (1.1). The simply nonlocal boundary value problem was presented and investigated for the first time by Bitsadze and Samarskii [12]. Further, methods of solutions of Bitsadze-Samarskii nonlocal boundary value problems for elliptic differential equations have been studied extensively by many researchers (see [13–21] and the references given therein).A functionu(t) is called a solution of problem (1.1) if the following conditions are satisfied.(i)
u(t) is twice continuously differentiable on the segment [0,1]. Derivatives at the endpoints of the segmentare understood asthe appropriate unilaterial derivatives.(ii)
The elementu(t) belongs to D(A) for all t∈[0,1], and the function Au(t) is continuous on [0,1].(iii)
u(t) satisfies the equation and nonlocal boundary condition in (1.1).LetΩ be the open unit cube in Rn(x=(x1,…,xn):0<xk<1,1≤k≤n) with boundary S,Ω¯=Ω∪S. In present paper, we are interested in studying the stable difference schemes for the numerical solution of the following nonlocal boundary value problem for the multidimensional elliptic equation
(1.2)-utt-∑r=1n‍(ar(x)uxr)xr+δu=f(t,x),0<t<1,x=(x1,…,xn)∈Ω,ut(0,x)=φ(x),ut(1,x)=βut(λ,x)+ψ(x),x∈Ω¯,|β|≤1,0≤λ<1,u(t,x)=0,0≤t≤1,x∈S,S=∂Ω¯.
Here ψ(x),φ(x)(x∈Ω¯), and f(t,x)(t∈(0,1),x∈Ω) are given smooth functions, δ is a large positive constant and ar(x)≥a>0.In the present paper, the first and second-orders of accuracy difference schemes are presented for the approximate solution of problem (1.2). The stability and coercive stability estimates for the solution of these difference schemes are obtained. A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichlet condition. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of two-dimensional elliptic partial differential equations.


## 2. Difference Schemes: Well-Posedness

The discretization of problem (1.2) is carried out in two steps. In the first step let us define the grid sets as follows:
(2.1)Ω~h={x:x=xm=(h1m1,…,hnmn),m=(m1,…,mn),0≤mr≤Nr,hrNr=L,r=1,…,n}Ωh=Ω~h∩Ω,Sh=Ω~h∩S.
We introduce the Hilbert space L2h=L2(Ω~h) of the grid functions φh(x)={φ(h1m1,…,hnmn)} defined on Ω~h, equipped with the norm
(2.2)∥φh∥L2(Ω~h)=(∑x∈Ωh¯‍|φh(x)|2h1⋯hn)1/2
and the Hilbert space W22(Ω~h) defined on Ω~h, equipped with the norm
(2.3)∥φh∥W22(Ω~h)=(∑x∈Ω~h‍|φh(x)|2h1⋯hn)1/2+(∑x∈Ω~h‍∑r=1n‍|φxr,mrh|2h1⋯hn)1/2+(∑x∈Ω~h‍∑r=1n‍|φxrx¯r,mrh|2h1⋯hn)1/2.
Finally, we introduce the Banach spaces C([0,1]τ,L2h) and Cα([0,1]τ,L2h) of grid abstract function {φkh(x)}1N-1 defined on [0,1]τ with values L2h, equipped with the following norms:
(2.4)∥{φkh}1N-1∥C([0,1]τ,L2h)=max1≤k≤N-1∥φkh∥L2h,∥{φkh}1N-1∥Cα([0,1]τ,L2h)=max1≤k≤N-1∥φkh∥L2h+sup1≤k<k+r≤N-1∥φk+r-φk∥L2h(rτ)α.To the differential operatorA generated by the problem (1.2) we assign the difference operator Ahx by the formula
(2.5)Ahxuh=-∑r=1n‍(ar(x)ux-rh)xr,mr+δuxrh
acting in the space of grid functions uh(x), satisfying the condition uh(x)=0 for all x∈Sh. It is known that Ahx is a self-adjoint positive definite operator in L2(Ω~h). With the help of Ahx, we arrive at the nonlocal boundary value problem for an infinite system of the following ordinary differential equations:
(2.6)-d2uh(t,x)dt2+Ahxuh(t,x)=fh(t,x),0<t<1,x∈Ωh,uth(0,x)=φh(x),uth(1,x)=βuth(λ,x)+ψh(x),x∈Ω~h.
In the second step, we replaced problem (2.6) by the first-order of accuracy difference scheme as follows:
(2.7)-uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)=fkh(x),fkh(x)=fh(tk,x),tk=kτ,1≤k≤N-1,Nτ=1,x∈Ωh,u1h(x)-u0h(x)τ=φh(x),x∈Ω~h,uNh(x)-uN-1h(x)τ=βu[λ/τ]+1h(x)-u[λ/τ]h(x)τ+ψh(x),x∈Ω~h
and the second order of accuracy difference scheme as follows:
(2.8)-uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)=fkh(x),fkh(x)=fh(tk,x),tk=kτ,1≤k≤N-1,Nτ=1,x∈Ωh,-3u0h(x)+4u1h(x)-u2h(x)2τ=φh(x),x∈Ω~h,uN-2h(x)-4uN-1h(x)+3uNh(x)2τ=β[u[λ/τ]-1h(x)-4u[λ/τ]h(x)+3u[λ/τ]+1h(x)2τ+u[λ/τ]+1h(x)-2u[λ/τ]h(x)+u[λ/τ]-1h(x)τ(λτ-[λτ])]+ψh(x),x∈Ω~h.
Now, we will study the well-posedness of (2.7) and (2.8). We have the following theorem on stability of (2.7) and (2.8).Theorem 2.1.
Letτ and |h| be sufficiently small positive numbers. Then the solutions of difference schemes (2.7) and (2.8) satisfy the following stability estimate:
(2.9)max1≤k≤N∥ukh∥L2h≤M1[∥φh∥L2h+∥ψh∥L2h+max1≤k≤N-1∥fkh∥L2h],
where M1 does not depend on τ,h,ψh(x),φh(x) and fkh(x),1≤k≤N-1.Proof.
The proof of (2.9) is based on the following formula:
(2.10)ukh(x)=(I-R2N)-1{∑i=1N-1‍(Rk-R2N-k)u0h(x)+(RN-k-RN+k)uNh(x)-(RN-k-RN+k)(2I+τBhx)-1(Bhx)-1∑i=1N-1‍(RN-1-i-RN-1+i)fih(x)τ}+(2I+τBhx)-1(Bhx)-1∑i=1N-1‍(R|k-i|-1-Rk+i-1)fih(x)τfork=1,…,N-1,
where(2.11)u0h(x)=Pτ(I+τBhx)(2I+τBhx)-1(Bhx)-1×[(I+R)RN-2∑i=1N-1‍(RN-i-RN+i)fih(x)τ+(I+R)[I+R2N-2-β(RN-[λ/τ]-1+RN+[λ/τ])]×∑i=1N-1‍Ri-1fih(x)τ-β(RN+RN-1)∑i=1[λ/τ]-1‍R[λ/τ]-ifih(x)τ+β(RN-2+RN-1)∑i=[λ/τ]+1N-1‍Ri-[λ/τ]fih(x)τ+β(RN+RN-1)∑i=1N-1‍R[λ/τ]+ifih(x)τ-β(RN-1+RN)f[λ/τ]h(x)τ]-Pτ(I-R)-1[I+R2N-1-β(RN-[λ/τ]-1+RN+[λ/τ])]φh(x)τ+Pτ(I-R)-1(RN-1+RN)ψh(x)τ,uNh(x)=Pτ(I+τBhx)(2I+τBhx)-1(Bhx)-1×[[R-1(I+R)-β(RN-[λ/τ]-1+RN+[λ/τ]-1)]∑i=1N-1‍(RN-i-RN+i)fih(x)τ-β(I+R2N-1)∑i=1[λ/τ]-1‍R[λ/τ]-ifih(x)τ+β(I+R2N-1)R-1∑i=[λ/τ]+1N-1‍Ri-[λ/τ]fih(x)τ+β(I+R2N-1)∑i=1N-1‍R[λ/τ]+ifih(x)τ-β(I+R2N-1)f[λ/τ]h(x)τ+(I+R)[RN+RN-1-β(R[λ/τ]+R2N-[λ/τ]-1)]∑i=1N-1‍Ri-1fih(x)τ]-Pτ(I-R)-1[RN+R2N-1-β(R[λ/τ]+R2N-[λ/τ]-1)]φh(x)τ+Pτ(I-R)-1(I+R2N-1)ψh(x)τ,Pτ=[I-R2N-2-β(RN-[λ/τ]-1+RN+[λ/τ]-1)]-1,
for (2.7), and
(2.12)u0h(x)=Dτ(I+τBhx)(2I+τBhx)-1(Bhx)-1×{(I+R)(4R-I-R2)(I-3R)RN-4(I-R2N)∑i=1N-2(RN-i-RN+i)fih(x)τ-(I+R)(4R-I-R2)(I-R2N)×[(3I-R+2(λτ-[λτ])(I-R))3I-R-R2N-2(I-3R)-β[RN-[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))-RN+[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))]]×∑i=2N-1Ri-2fih(x)τ-β(I+R)RN-2(4R-I-R2)(I-R2N)×[(I-3R+2(λτ-[λτ])(I-R))∑i=1[λ/τ]-1R[λ/τ]-i-1fih(x)τ+(3I-R+2(λτ-[λτ])(I-R))∑i=[λ/τ]+2N-1Ri-[λ/τ]-1fih(x)τ-(I-3R+2(λτ-[λτ])(I-R))∑i=1N-1R[λ/τ]+i-1fih(x)τ]-β(I-R2N)(I+R)RN-2(4R-R2-I)(4+4(λτ-[λτ]))×(f[λ/τ]+1h(x)τ-f[λ/τ]h(x)τ)-(I+R)(4I-R)(I-R2N)×[(I-3R+2(λτ-[λτ])(I-R))3I-R-R2N-2(I-3R)-β(RN-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))-RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R)))]×f1h(x)τ-(I+R)RN-1(4R-R2-I)(I-R2N)(4I+R2N-3(I-3R))×fN-1h(x)τ(∑i=1N-2(RN-i-RN+i)fih(x)τ)}-Dτ(I-R2N)(I-R)-1(I+R)RN-2(4R-I-R2)2τψh(x)+Dτ(I-R2N)(I-R)-1×[(λτ-[λτ])-RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))3I-R-R2N-2(I-3R)-β[RN-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))-RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))]]2τφh(x),(2.13)uNh(x)=Dτ(I+τBhx)(2I+τBhx)-1(Bhx)-1×{RN+[λ/τ]-1(I-R2N)[(I-3R+2(λτ-[λτ])(I-R))(I+R)(4R-I-R2)R-2(R-3I)+β[(R-3I)(3I-R+2(λτ-[λτ])(I-R))RN-[λ/τ]-1-(3R-I)(I-3R+2(λτ-[λτ])(I-R))RN+[λ/τ]-1]]×∑i=1N-2(RN-i-RN+i)fih(x)τ+(I+R)(I-R2N)(4R-I-R2)×[-R2N-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))](I+R)RN-2(4R-I-R2)-β[R[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))-R2N-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))]]∑i=2N-1Ri-2fih(x)τ+β(I-R2N)(R-3I+R2N-2(I-3R))×[(I-3R+2(λτ-[λτ])(I-R))∑i=1[λ/τ]-1R[λ/τ]-i-1fih(x)τ+(3I-R+2(λτ-[λτ])(I-R))∑i=[λ/τ]+2N-1Ri-[λ/τ]-1fih(x)τ-(I-3R+2(λτ-[λτ])(I-R))∑i=1N-1R[λ/τ]+i-1fih(x)τ]+β(R-3I+R2N-2(3I-R))(I-R2N)×(4+4(λτ-[λτ])(I-R))(f[λ/τ]+1h(x)τ-f[λ/τ]h(x)τ)+(I+R)(I-R2N)×[-R2N-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))](I+R)RN-2(I-4R+R2)-β[R[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))-R2N-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))]]f1h(x)τ+R(I-R2N-1)[RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))R-4I+R2N-4(I+R2-3R)(3I-R-R2(4I-R))-β[RN-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))×(3R-I+R2N(3I-R))-RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))×(3I-R+R2N(I-3R))-RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))]]fN-1h(x)τ-RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))(3I-R+R2N(I-3R))]]}+Dτ(I-R2N)(I-R)-1(R-3I+R2N(I-3R))2τφh(x)-Dτ(I-R2N)(I-R)-1×[(3I-R+2(λτ-[λτ])(I-R))RN-2(I+R)(R2-4R+I)-β[R[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))-R2N-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))]]2τψh(x),Dτ=(I-R2N)-1×{(3I-R+2(λτ-[λτ])(I-R))[-(3I-R)2+(I-3R)2]-β[-RN+[λ/τ]-3(I-3R)(I-3R+2(λτ-[λτ])(I-R))-RN-[λ/τ]-1(3I-R)(3I-R+2(λτ-[λτ])(I-R))]}-1,R=(I+τBhx)-1,Bhx=12(τAhx+4Ahx+τ2(Ahx)2)
for (2.8), and the symmetry properties of the difference operator Ahx defined by the formula (2.5).Difference schemes (2.7) and (2.8) are ill-posed in C([0,1]τ,L2h). We have the following theorem on almost coercive stability.Theorem 2.2.
Letτ and |h| be sufficiently small positive numbers. Then the solutions of difference schemes (2.7) and (2.8) satisfy the following almost coercive stability estimate:
(2.14)max1≤k≤N-1∥uk+1h-2ukh+uk-1hτ2∥L2h+max1≤k≤N-1∥ukh∥W2h2≤M2[∥φh∥W2h2+∥ψh∥W2h2+ln1τ+|h|max1≤k≤N-1∥fkh∥L2h],
where M2 does not depend on τ,h,ψh(x),φh(x), and fkh(x),1≤k≤N-1.Proof.
The proof of (2.14) is based on the formulas (2.11), (2.12), (2.13), the symmetry properties of the difference operator Ahx defined by the formula (2.5) and on the following theorem on well-posedness of the elliptic difference problem.Theorem 2.3.
For the solutions of the elliptic difference problem(2.15)Ahxuh(x)=wh(x),x∈Ωh,uh(x)=0,x∈Sh
the following coercivity inequality holds (see [21]):
(2.16)∑r=1n∥(uh)xrxr-,mr∥L2h≤M∥wh∥L2h,
where M  does not depend on h and wh(x).Theorem 2.4.
Letφh(x)=ψh(x)=0. Then the difference problems (2.7) and (2.8) are well-posed in Hölder spaces Cα([0,1]τ,L2h) and the following coercivity inequality holds:
(2.17)max1≤k≤N-1∥{uk+1h-2ukh+uk-1hτ2}1N-1∥Cα([0,1]τ,L2h)+∥{ukh}1N-1∥Cα([0,1]τ,W2h2)≤M3α(1-α)max1≤k≤N-1∥{fkh}1N-1∥Cα([0,1]τ,L2h).
Here M3 does not depend on τ,h, and fkh(x),1≤k≤N-1.Proof.
The proof of (2.17) is based on the formula
(2.18)Ahxu0h(x)-f1h(x)=Pτ(I+τBhx)(2I+τBhx)-1(I+R)×[RN-1∑i=1N-1‍τBhxRN-i(fih-fN-1h)-RN-1∑i=1N-1‍τBhxRN+i(fih(x)-f1h(x))-βRN∑i=1[λ/τ]-1‍τBhxR[λ/τ]-i(fih(x)-f[λ/τ]h(x))+[I+R2N-2-β(RN-[λ/τ]-1+RN+[λ/τ])]×∑i=1N-1‍τBhxRi(fih(x)-f1h(x))+βRN∑i=[λ/τ]+1N-1‍τBhxRi-[λ/τ](fih(x)-f[λ/τ]h(x))+βRN∑i=1N-1‍τBhxR[λ/τ]+i(fih(x)-f1h(x))+RN-1(I-RN-1)fN-1h(x)+β[RN+[λ/τ]-1-R2N-[λ/τ]-1-τBhxRN]f[λ/τ]h(x)+[+β(R2N-[λ/τ]-2+R2N+[λ/τ]-1+R2N+[λ/τ]-RN+[λ/τ]-1)∑i=1N-1‍]R2N-2-R2N-1+R2N-2-R3N-3+R3N-2-RN-1+β(R2N-[λ/τ]-2+R2N+[λ/τ]-1+R2N+[λ/τ]-RN+[λ/τ]-1)∑i=1N-1‍]f1h(x)]-Pτ(I-R)-1τ[I+R2N-1-β(RN-[λ/τ]-1+RN+[λ/τ])]Ahxφh+Pτ(I-R)-1τ(RN-1+RN)Ahxψh,AxhuNh(x)-fN-1h(x)=Pτ(I+τBhx)(2I+τBhx)-1×{∑i=1N-1‍τBhxRN-i(fih(x)-fN-1h(x))[R(I+R)-β(RN-[λ/τ]+1+RN+[λ/τ]+1)]×∑i=1N-1‍τBhxRN-i(fih(x)-fN-1h(x))-[R(I+R)-β(RN-[λ/τ]+1+RN+[λ/τ]+1)]×∑i=1N-1‍τBhxRN+i(fih(x)-f1h(x))-βR2(I+R2N-1)∑i=1[λ/τ]-1‍τBhxR[λ/τ]-i(fih(x)-f[λ/τ]h(x))+βR(I+R2N-1)∑i=[λ/τ]+1N-1‍τBhxRi-[λ/τ](fih(x)-f[λ/τ]h(x))+βR2(I+R2N-1)∑i=1N-1‍τBhxR[λ/τ]+i(fih(x)-f1h(x))-β(I+R2N-1)×[R(I-R[λ/τ]-1)-(I-RN-[λ/τ]-1)-τBhxR]×f[λ/τ]h(x)+[(I+R)(R2N-2-RN-I+R)-β(RN-[λ/τ]+1+RN+[λ/τ]+1-R2N-[λ/τ]-R2N+[λ/τ]-RN-[λ/τ]-1+RN+[λ/τ]-1-RN-[λ/τ]+RN+[λ/τ])]×fN-1h(x)+[(I+R)(RN-R2N-1)+β(RN+[λ/τ]+RN+[λ/τ]+1-RN+[λ/τ]+2+R2N+[λ/τ]+1+R2N+[λ/τ]+1-R3N+[λ/τ]-R3N+[λ/τ]+1+R[λ/τ]+1-R2N-[λ/τ]+R3N-[λ/τ]-1)]f1h(x)∑i=1N-1‍}+Pτ(I-R)-1Rτ(I+R2N-1)Axhψ-Pτ(I-R)-1Rτ[RN-1+RN-β(R[λ/τ]+R2N-[λ/τ]-1)]Axhφ
for (2.7) and
(2.19)Ahxu0h(x)-f1h(x)=Dτ(I+τBhx)(2I+τBhx)-1×{∑i=1N-2BhxτRN-i(fih(x)-fN-1h(x))-(I+R)(4R-I-R2)(I-3R)RN-3(I-R2N)×[∑i=1N-2BhxτRN-i(fih(x)-fN-1h(x))-∑i=1N-2BhxτRN+i(fih(x)-f1h(x))]-(I+R)(4R-I-R2)(I-R2N)×[(I-3R+2(λτ-[λτ])(I-R))3I-R-R2N-2(I-3R)-β[RN-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))-RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))]]×∑i=2N-1BhxτRi-1(fih(x)-f1h(x))-β(I+R)RN-2(4R-I-R2)(I-R2N)×[∑i=1N-1BhxτR[λ/τ]+i(fih(x)-f1h(x))(I-3R+2(λτ-[λτ])(I-R))×∑i=1[λ/τ]-1BhxτR[λ/τ]-i(fih(x)-f[λ/τ]h(x))+(3I-R+2(λτ-[λτ])(I-R))×∑i=[λ/τ]+2N-1BhxτRi-[λ/τ](fih(x)-f[λ/τ]h(x))-(I-3R+2(λτ-[λτ])(I-R))×∑i=1N-1BhxτR[λ/τ]+i(fih(x)-f1h(x))]-β(I-R2N)(I+R)RN-1(4R-R2-I)×(4+4(λτ-[λτ]))τBhxf[λ/τ]+1h(x)+β(I-R2N)(I+R)RN-2(4R-R2-I)×[2R(R+2(λτ-[λτ]))+R[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))+RN-[λ/τ]-2×(3I-R+2(λτ-[λτ])(I-R))]f[λ/τ]h(x)×[-(I-3R+2(λτ-[λτ])(I-R))RN+[λ/τ]-3)](I+R)(I-R2N)×[(I-3R+2(λτ-[λτ])(I-R))(I-3R)(3R-I-R2N-5(I-R)×(R2-R2N(I+R)(R2-4R+I)))-(3I-R)RN-1(R2-4R+I)-β(4R-I-R2)×((3I-R+2(λτ-[λτ])(I-R))RN-[λ/τ]-3-(I-3R+2(λτ-[λτ])(I-R))×RN+[λ/τ]-3)(I-3R+2(λτ-[λτ])(I-R))]]f1h(x)+(I+R)(4R-I-R2)(I-R2N)RN-2×(R-3I-RN-2(I-3R)(I+RN(I-R)))fN-1h(x)∑i=1N-2BhxτRN-i(fih(x)-fN-1h(x))-}-Dτ(I-R2N)(I-R)-1(I+R)RN-2(4R-I-R2)2τAhxψh(x)+Dτ(I-R2N)(I-R)-1×[(I-3R+2(λτ-[λτ])(I-R))3I-R-R2N-2(I-3R)-β[RN-[λ/τ](3I-R+2(λτ-[λτ])(I-R))-RN+[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))]]2τAhxφh(x),AhxRuNh(x)-fN-1h(x)=Dτ(I+τBhx)(2I+τBhx)-1×{-(3R-I)(I-3R+2(λτ-[λτ])(I-R))RN+[λ/τ]+1](I-R2N)×[-(3R-I)(I-3R+2(λτ-[λτ])(I-R))RN+[λ/τ]+1](I+R)(4R-I-R2)(R-3I)+β[(R-3I)(3I-R+2(λτ-[λτ])(I-R))×RN-[λ/τ]+1-(3R-I)×(I-3R+2(λτ-[λτ])(I-R))RN+[λ/τ]+1]]×(∑i=1N-2BhxτRN-i(fih(x)-fN-1h(x))-∑i=1N-2BhxτRN+i(fih(x)-f1h(x)))+(I+R)(I-R2N)(4R-I-R2)×[(I-3R+2(λτ-[λτ])(I-R))(I+R)RN-2(-4R+I+R2)-β[R[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))-R2N-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))]]×∑i=2N-1BhxτRi(fih(x)-f1h(x))+βR(R-3I+R2N-2(I-3R))(I-R2N)×[(I-3R+2(λτ-[λτ])(I-R))×∑i=1[λ/τ]-1BhxτR[λ/τ]-i(fih(x)-f[λ/τ]h(x))+(3I-R+2(λτ-[λτ])(I-R))×∑i=[λ/τ]+2N-1BhxτR[λ/τ]+i(fih(x)-f1h(x))+(I-3R+2(λτ-[λτ])(I-R))×∑i=1N-1BhxτR[λ/τ]+i(fih(x)-f1h(x))]+βR2(R-3I+R2N-2(I-3R))(I-R2N)(4+4(λτ-[λτ]))×f[λ/τ]-1h(x)-βR(I-R2N)×[(I-3R+2(λτ-[λτ])(I-R))(I-R+RN-[λ/τ])(3I-R+2(λτ-[λτ])(I-R))+R[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))]×f[λ/τ]h(x)+(I-R2N)×[(3I-R+2(λτ-[λτ])(I-R))(I+R)(4R-R2-I)RN+1×(R-3I)(I-RN-2)+β[(3I-R+2(λτ-[λτ])(I-R))×R2N-[λ/τ](R2(3I-R)+RN-2(R3-4R2+I))-(I-3R+2(λτ-[λτ])(I-R))R[λ/τ]×(R2(3I-R)+(I-3R)RN-3(R+RN(I-R)))(3I-R+2(λτ-[λτ])(I-R))]]×f1h(x)+(I-R2N)×[(3I-R+2(λτ-[λτ])(I-R))(I-3R)(4R-R2-I)(3I-R+R2N-2)-β[(3I-R+2(λτ-[λτ])(I-R))RN-[λ/τ]-1(3I-R)×(R+I+RN(I-R+RN-2(I-R)+R3N-2(2I-R)))-(I-3R+2(λτ-[λτ])(I-R))RN+[λ/τ]-1(3R-I)×(I-R-R2N-3(3I-R+2(λτ-[λτ])(I-R))×(1+R-RN(3I-R2)+R2N(R3-4R2+I)))]]×fN-1h(x)×(I-R-R2N-3(1+R-RN(3I-R2)+R2N(R3-4R2+I)))(3I-R+2(λτ-[λτ])(I-R))]]}+Dτ(I-R2N)(I-R)-1R(R-3I+R2N-2(I-3R))2τAhxφh(x)-Dτ(I-R2N)(I-R)-1R×[-R2N-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))]RN-2(I+R)(R2-4R+I)-β[R[λ/τ]-1(I-3R+2(λτ-[λτ])(I-R))-R2N-[λ/τ]-1(3I-R+2(λτ-[λτ])(I-R))]]2τAhxψh(x)
for (2.8), the symmetry properties of the difference operator Ahx defined by the formula (2.5) and on Theorem 2.3 on well-posedness of the elliptic difference problem.


## 3. Numerical Analysis

We have not been able to obtain a sharp estimate for the constants figuring in the stability inequality. Therefore, we will give the following results of numerical experiments of the Bitsadze-Samarskii-Dirichlet problem:(3.1)-∂2u(t,x)∂t2-∂2u(t,x)∂x2+u=f(t,x),0<t<1,0<x<1,f(t,x)=exp(-πt)sin(πx),0<t<1,0<x<1,ut(0,x)=-πsin(πx),0≤x≤1,ut(1,x)=ut(12,x)+πsin(πx)(exp(-π2)-exp(-π)),0≤x≤1,u(t,0)=u(t,1)=0,0≤t≤1
for the two-dimensional elliptic equation.The exact solution of this problem is(3.2)u(t,x)=exp(-πt)sin(πx).
For the approximate solution of problem (3.1), we consider the set [0,1]τ×[0,1]h of a family of grid points depending on small parameters τ and h as follows:
(3.3)[0,1]τ×[0,1]h={(tk,xn):tk=kτ,0≤k≤N,Nτ=1,xn=nh,0≤n≤M,Mh=1}.Applying (2.7), we present the following first-order of accuracy difference scheme for the approximate solution of problem (3.1):
(3.4)-uk+1n-2ukn+uk-1nτ2-ukn+1-2ukn+ukn-1h2+ukn=exp(-πtk)sin(πxn),1≤k≤N-1,1≤n≤M-1,u1n-u0nτ=-πsin(πxn),0≤n≤M,uNn-uN-1nτ=βuN/2n-uN/2-1nτ+πsin(πxn)×(exp(-π2)-exp(-π)),0≤n≤M,uk0=ukM=0,0≤k≤N.We have(N+1)×(M+1) system of linear equations in (3.4) and we will write them in the following matrix form:
(3.5)Aun+1+Bun+Cun-1=Dφn,1≤n≤M-1,u0=uM=0~.
Here,
(3.6)A=[000·00·0000a0·00·00000a·00·000··········000·00·a00000·00·0a0000·00·000](N+1)×(N+1),B=[-110·00·000cbc·00·0000cb·00·000··········000·00·bc0000·00·cbc000·1-1·0-11](N+1)×(N+1),C=A,D=[I](N+1)×(N+1),us=[us0us1·usN-1usN](N+1)×1,
wheres=n-1,n,n+1 and
(3.7)φn=[φn0φn1.φnN-1φnN](N+1)×1,a=1h2,b=-2τ2-2h2-1,c=1τ2,φnk={-τπsin(πxn),k=0,-exp(-πtk)sin(πxn),1≤k≤N-1,τπsin(πxn)(exp(-π2)-exp(-π)),k=N.
We seek a solution of the matrix equation in following form:
(3.8)un=αn+1un+1+βn+1,n=M-1,…,1,uM=0~,
where αn(n=1,…,M) are (N+1)×(N+1) square matrices and βn(n=1,…,M) are (N+1)×1 column matrices defined by (see, [17]) as follows:
(3.9)αn+1=-(B+Cαn)-1A,βn+1=(B+Cαn)-1(Dφn-Cβn),n=1,…,M-1,
where
(3.10)α1=[0](N+1)×(N+1),β1=[0](N+1)×1.Now, applying (2.8) for N even number, we can present the following second-order of accuracy difference scheme:
(3.11)-uk+1n-2ukn+uk-1nτ2-ukn+1-2ukn+ukn-1h2+ukn=-exp(-πtk)sin(πxn),1≤k≤N-1,1≤h≤M-1,-3u0n+4u1n-u2n2τ=-πsin(πxn),0≤n≤M,uN-2n(x)-4uN-1n(x)+3uNn(x)2τ=uN/2-2n-4uN/2-1n+3uN/2n2τ+πsin(πxn)(exp(-π2)-exp(-π)),0≤n≤M,uk0=ukM=0,0≤k≤N
for the approximate solution of problem (3.1).So, again we have(N+1)×(M+1) system of linear equations in (3.11) and we will write them in the following matrix form:
(3.12)Aun+1+Bun+Cun-1=Dφn,1≤n≤M-1,u0=uM=0~,
where
(3.13)A=[000·000·0000a0·000·00000a·000·000···········000·000·a00000·000·0a0000·000·000](N+1)×(N+1),B=[-34-1·000·000bcb·000·0000bc·000·000···········000·000·cb0000·000·bcb000·-14-3·1-43](N+1)×(N+1),C=A,D=[I](N+1)×(N+1),us=[us0us1·usN-1usN](N+1)×1,
where s=n-1,n,n+1 and
(3.14)φnk=[φn0φn1·φnN-1φnN](N+1)×1.
Here,
(3.15)a=1h2,b=1τ2,c=-2h2-2τ2-1,(3.16)φnk={-2τπsin(πxn),k=0,-exp(-πtk)sin(πxn),1≤k≤N-1,2τπsin(πxn)(exp(-π2)-exp(-π)),k=N.
So, we have the second-order difference equation with respect to n with matrix coefficients. To solve this difference equation, we use the same algorithm (3.8) and (3.9).Now, we will give the results of the numerical experiments.The errors in numerical solutions are computed by(3.17)EMN=max1≤k≤N-1(∑n=1M-1‍|u(tk,xn)-unk|2h)1/2
for different values of M and N, where u(tk,xn) represents the exact solution and unk represents the numerical solution at (tk,xn). The results are shown in Table  1 for N=M=20, N=M=40, and N=M=60.Table 1
Comparison of the errors of difference schemes.
Difference schemes
N
=
M
=
20
N
=
M
=
40
N
=
M
=
60
First-order difference scheme (2.7)
0.05384197882300
0.02631633782987
0.01740639813932
Second-order difference scheme (2.8)
0.00631619894867
0.00164455475890
7.4144589892985e-004Thus, second-order of accuracy difference scheme is more accurate compared with the first-order of accuracy difference scheme.


## 4. Conclusion

The first and second-orders of accuracy difference schemes for approximate solutions of the Bitsadze-Samarskii-Dirichlet type nonlocal boundary value problem for the multidimensional elliptic partial differential equation are presented. Theorems on the stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are established. Numerical experiments are given.



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*Source: 454831-2012-11-05.xml*