We are interested in studying the stable difference schemes for the numerical solution of
the nonlocal boundary value problem with the Dirichlet-Neumann condition for the multidimensional elliptic equation. The first and second orders of accuracy difference schemes
are presented. A procedure of modified Gauss elimination method is used for solving these
difference schemes for the two-dimensional elliptic differential equation. The method is
illustrated by numerical examples.

1. Introduction

Methods of solution of the Bitsadze-Samarskii nonlocal boundary value problems for elliptic differential equations have been studied extensively by many researchers (see [1–22] and the references given therein).

Let Ω be the unit open cube in Rm(x=(x1,…,xm):0<xk<1,1≤k≤m) with boundary S,Ω¯=Ω∪S. In [0,1]×Ω, the Bitsadze-Samarskii-type nonlocal boundary value problem for the multidimensional elliptic equation
(1.1)-utt-∑r=1m(ar(x)uxr)xr+ηu=f(t,x),0<t<1,x=(x1,…,xm)∈Ω,u(0,x)=φ(x),u(1,x)=∑j=1Jαju(λj,x)+ψ(x),x∈Ω¯,∑j=1J|αj|≤1,0<λ1<⋯<λJ<1,u(t,x)∣x∈S1=0,∂u(t,x)∂n→|x∈S2=0,S1∪S2=S
is considered. Here ar(x),(x∈Ω),ψ(x),φ(x)(x∈Ω¯), and f(t,x)(t∈(0,1),x∈Ω) are given smooth functions, ar(x)≥a>0, η is a positive number, and n→ is the normal vector to Ω. We are interested in studying the stable difference schemes for the numerical solution of the nonlocal boundary value problem (1). The first and second orders of accuracy difference schemes are presented. The stability and almost coercive stability of these difference schemes are established. 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: The Stability and Coercive Stability Estimates

The discretization of problem (1) is carried out in two steps. In the first step, let us define the grid sets
(2.1)Ω~h={x=xm=(h1m1,…,hmmm),m=(m1,…,mm),0≤mr≤Nr,hrNr=1,r=1,…,m},Ωh=Ω~h∩Ω,Shr=Ω~h∩Sr,r=1,2.
We introduce the Hilbert space L2h=L2(Ω~h) and W2h2=W22(Ω~h) of the grid functions φh(x)={φ(h1m1,…,hmmm)} defined on Ω~h, equipped with the norms
(2.2)‖φh‖W2h2=‖φh‖L2h+(∑x∈Ωh¯∑r=1m|(φh)xr|2h1⋯hm)1/2+(∑x∈Ωh¯∑r=1m|(φh)xrxr¯,mr|2h1⋯hm)1/2,‖φh‖L2(Ω~h)=(∑x∈Ωh¯|φh(x)|2h1⋯hm)1/2.
To the differential operator A generated by problem (1), we assign the difference operator Ahx by the formula
(2.3)Ahxuh=-∑r=1m(ar(x)uxr-h)xr,jr+ηuh(x),
acting in the space of grid functions uh(x), satisfying the conditions uh(x)=0 for all x∈Sh1 and Dhuh(x)=0 for all x∈Sh2. Here, Dhuh(x) is an approximation to ∂u/∂n→. 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
(2.4)-d2uh(t,x)dt2+Ahxuh(t,x)=fh(t,x),0<t<1,x∈Ωh,uh(0,x)=φh(x),uh(1,x)=∑j=1Jαjuh(λj,x)+ψh(x),x∈Ω~h,∑j=1J|αj|≤1,0<λ1<⋯<λJ<1,
for an infinite system of ordinary differential equations. In the second step, we replace problem (2.4) by the first and second orders of accuracy difference schemes
(2.5)-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,u0h(x)=φh(x),x∈Ω~h,uNh(x)=∑j=1Jαju[λj/τ]h(x)+ψh(x),x∈Ω~h,(2.6)-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,u0h(x)=φh(x),x∈Ω~h,uNh(x)=∑j=1Jαj(u[λj/τ]h(x)+(u[λj/τ]+1h(x)-u[λj/τ]h(x))(λjτ-[λjτ]))+ψh(x),x∈Ω~h.
To formulate our result on well-posedness, we will give definition of C01α([0,1]τ,H) and C([0,1]τ,H). Let F([0,1]τ,H) be the linear space of mesh functions φτ={φk}1N-1 with values in the Hilbert space H. We denote C([0,1]τ,H) normed space with the norm
(2.7)‖φτ‖C([0,1]τ,H)=max1≤k≤N-1‖φk‖H,
and C01α([0,1]τ,H) normed space with the norm
(2.8)‖φτ‖C01α([0,1]τ,H)=‖φτ‖C([0,1]τ,H)+sup1≤k≤k+r≤N-1((N-k)τ)α((k+r)τ)α(rτ)α‖φk+r-φk‖H.

Theorem 2.1.

Let τ and |h| be sufficiently small positive numbers. Then, the solutions of difference schemes (2.5) and (2.6) satisfy the following stability and almost coercive stability estimates
(2.9)‖{ukh}1N-1‖C([0,1]τ,L2h)≤M1[‖φh‖L2h+‖ψh‖L2h+‖{fkh}1N-1‖C([0,1]τ,L2h)],‖{uk+1h-2ukh+uk-1hτ2}1N-1‖C([0,1]τ,L2h)+‖{ukh}1N-1‖C([0,1]τ,W2h2)≤M2[‖φh‖W2h2+‖ψh‖W2h2+ln1τ+|h|‖{fkh}1N-1‖C([0,1]τ,L2h)].
Here, M1 and M2 do not depend onτ,h,ψh(x),φh(x), and fkh(x),1≤k≤N-1.

Theorem 2.2.

Let τ and |h| be sufficiently small positive numbers. Then, the solution of difference schemes (2.5) and (2.6) satisfies the following coercive stability estimate:
(2.10)‖{uk+1h-2ukh+uk-1hτ2}1N-1‖C01α([0,1]τ,L2h)+‖{ukh}1N-1‖C01α([0,1]τ,W2h2)≤M3[‖φh‖W2h2+‖ψh‖W2h2+1α(1-α)‖{fkh}1N-1‖C01α([0,1]τ,L2h)].M3 is independent of τ,h,ψh(x),φh(x), and fkh(x),1≤k≤N-1.

Proofs of Theorems 2.1 and 2.2 are based on the symmetry properties of operator Ahx defined by formula (2.3) and on the following formulas:
(2.11)ukh(x)=(I-R2N)-1×{∑i=1N-1(Rk-R2N-k)φh(x)+(RN-k-RN+k)uNh(x)-(RN-k-RN+k)×(I+τB)(2I+τB)-1B-1∑i=1N-1(RN-i-RN+i)fih(x)τ}+(I+τB)(2I+τB)-1B-1∑i=1N-1(R|k-i|-Rk+i)fih(x)τ,uNh(x)=D-1(∑k=1Jαk(I-R2N)-1×{∑i=1N-1(R[λk/τ]-R2N-[λk/τ])φh(x)-(RN-[λk/τ]-RN+[λk/τ])(I+τB)(2I+τB)-1B-1×∑i=1N-1(RN-i-RN+i)fih(x)τ}+(I+τB)(2I+τB)-1B-1×(∑i=1[λk/τ]R[λk/τ]-ifih(x)τ+∑i=[λk/τ]+1N-1Ri-[λk/τ]fih(x)τ-∑i=1N-1R[λk/τ]+ifih(x)τ)+ψh(x)),
for difference scheme (2.5), and
(2.12)uNh(x)=D-1(∑k=1Jαk(I-R2N)-1×{∑i=1N-1(R[λk/τ]-R2N-[λk/τ])φh(x)-(RN-[λk/τ]-RN+[λk/τ])(I+τB)(2I+τB)-1×B-1∑i=1N-1(RN-i-RN+i)fih(x)τ}+(I+τB)(2I+τB)-1B-1×(∑i=1[λk/τ]R[λk/τ]-ifih(x)τ+∑i=[λk/τ]+1N-1Ri-[λk/τ]fih(x)τ-∑i=1N-1R[λk/τ]+ifih(x)τ)+(λkτ-[λkτ])(I-R2N)-1×{∑i=1N-1τB(R[λk/τ]+1-R2N-[λk/τ])φh(x)-(RN-[λk/τ]-1-RN+[λk/τ])(2I+τB)-1×∑i=1N-1(RN-i-RN+i)fih(x)τ2}+(2I+τB)-1(∑i=1[λk/τ]R[λk/τ]-ifih(x)τ2+∑i=[λk/τ]+1N-1Ri-[λk/τ]-1fih(x)τ2-∑i=1N-1R[λk/τ]+ifih(x)τ2)+ψh(x)),
for difference scheme (2.6). Here,
(2.13)R=(I+τB)-1,B=τA2+τ2A24+A,A=Ahx,D=I-R2N-∑k=1Jαk(RN-[λk/τ]-RN+[λk/τ])for(2.5),D=I-R2N-∑k=1Jαk(RN-[λk/τ]-RN+[λk/τ]-1τ(λk-[λkτ]τ)×B(RN-[λk/τ]-RN+[λk/τ]+1)[λkτ])for(2.6),
and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h.

Theorem 2.3 (see [<xref ref-type="bibr" rid="B23">22</xref>]).

For the solution of the elliptic difference problem
(2.14)Ahxuh(x)=ωh(x),x∈Ω~h,uh(x)|x∈Sh1=0,Dhuh(x)|x∈Sh2=0,Sh1∪Sh2=Sh,
the following coercivity inequality holds:
(2.15)∑r=1m‖(uh)x¯rx¯r,jr‖L2h≤M4‖ωh‖L2h,
where M4 does not depend on h and ωh(x).

Note that we have not been able to obtain sharp estimate for the constants figuring in the stability estimates. Hence, in the following section, we study difference schemes (2.5) and (2.6) by numerical experiments.

3. Numerical Results

For the numerical result, we consider the nonlocal boundary value problem
(3.1)-∂2u(t,x)∂t2-∂2u(t,x)∂x2+u=2exp(-t)(x-12x2+t2-1),0<t<1,0<x<1,u(0,x)=x2-2x,u(1,x)=u(12,x)+(x22-x)exp(-1)-(3x24-3x2)exp(-12),0≤x≤1,u(t,0)=ux(t,1)=0,0≤t≤1,
for the elliptic equation. The exact solution of (3.1) is
(3.2)u(t,x)=(tx-tx22+x2-2x)exp(-t).
For the approximate solution of the nonlocal boundary Bitsadze-Samarskii problem (3.1), we consider the set [0,1]τ×[0,1]h of a family of grid points depending on the small parameters τ and h(3.3)[0,1]τ×[0,1]h={(tk,xn):tk=kτ,1≤k≤N-1,Nτ=1xn=nh,1≤n≤M-1,Mh=1}.
Firstly, applying difference scheme (2.5), we present the first order of accuracy difference scheme for the approximate solution of problem (3.1) is
(3.4)-unk+1-2unk+unk-1τ2-un+1k-2unk+un-1kh2+unk=f(tk,xn),1≤k≤N-1,1≤n≤M-1,un0=φ(xn),0≤n≤M,unN=un[N/2]+(xn22-xn)exp(-1)-(3xn24-3xn2)exp(-12),0≤n≤M,u0k=uMk-uM-1kh=0,0≤k≤N,f(tk,xn)=2exp(-tk)(xn-xn22+tk2-1),φ(xn)=xn2-2xn.
Then, we have an (N+1)×(M+1) system of linear equations and we will write them in the matrix form
(3.5)AUn+1+BUn+CUn-1=Dφn,1≤n≤M-1,U0=0~,UM-UM-1=0~,
where
(3.6)A=[000⋅0⋅0000a0⋅0⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅0⋅0a0000⋅0⋅000](N+1)×(N+1),B=[100⋅0⋅000cbc⋅0⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅0⋅cbc000⋅-1⋅001](N+1)×(N+1),
and C=A,D is an (N+1)×(N+1) identity matrix and
(3.7)Us=[us0us1⋅usN-1usN](N+1)×1,
where s=n-1,n,n+1,
(3.8)φn=[φn0φn1⋅φnN-1φnN](N+1)×1.
Here,
(3.9)a=-1h2,b=2τ2+2h2+1,c=-1τ2,φnk={(xn2-2xn),k=0,f(tk,xn),1≤k≤N-1,(xn22-xn)exp(-1)-(3xn24-3xn2)exp(-12),k=N.
So, we have a second-order difference equation with respect to n matrix coefficients. To solve this difference equation, we have applied a procedure of modified Gauss elimination method for difference equation with respect to n matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:
(3.10)Uj=αj+1Uj+1+βj+1,j=M-1,…,1,UM=(I-αM)-1βM,αj+1=-(B+Cαj)-1A,βj+1=(B+Cαj)-1(Dφj-Cβj),j=1,…,M-1,
where αj(j=1,…,M) are (N+1)×(N+1) square matrix and βj(j=1,…,M) are (N+1)×1 column matrix and α1 is the (N+1)×(N+1) zero matrix and βj is the (N+1)×1 zero matrix. Secondly, applying difference scheme (2.6), we present the following second order of accuracy difference scheme for the approximate solutions of problem (3.1):
(3.11)-unk+1-2unk+unk-1τ2-un+1k-2unk+un-1kh2+unk=f(tk,xn),1≤k≤N-1,1≤n≤M-1,un0=φ(xn),0≤n≤M,unN=un[N/2]+(un[N/2]+1-un[N/2])(N2-[N2])+(xn22-xn)exp(-1)-(3xn24-3xn2)exp(-12),0≤n≤M,u0k=0,uM-2k-4uM-1k+3uMk=0,0≤k≤N,f(tk,xn)=2exp(-tk)(xn-xn22+tk2-1),φ(xn)=xn2-2xn.
So, we have again an (N+1)×(M+1) system of linear equations and we will write in the matrix form
(3.12)AUn+1+BUn+CUn-1=Rφn,1≤n≤M-1,U0=0~,UM-2-4UM-1+3UM=0~,
where
(3.13)A=[000⋅00⋅0000a0⋅00⋅000⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅000⋅00⋅0a0000⋅00⋅000](N+1)×(N+1),B=[100·00·000cbc·00·000··········000·00·cbc000·de·001](N+1)×(N+1),C=A,R=D,Us=[us0us1⋅usN-1usN](N+1)×1,
where s=n-1,n,n+1 and φn=[φn0φn1·φnN-1φnN](N+1)×1.

Here,
(3.14)a=-1h2,b=2h2+2τ2+1,c=-1τ2,d=[N2]-N2,e=-1-d,φnk={(xn2-2xn),k=0,f(tk,xn),1≤k≤N-1,(xn22-xn)exp(-1)-(3xn24-3xn2)exp(-12),k=N.
Thus, we have a second-order difference equation with respect to n matrix coefficients. To solve this difference equation, we have applied the same procedure of modified Gauss elimination method (3.10) for difference equation with respect to n matrix coefficients with
(3.15)UM=(3I+αMαM-1-4αM)-1(-βMαM-1-βM-1+4βM).
Now, we will give the results of the numerical analysis. The errors computed by
(3.16)EMN=max1≤k≤N-1(∑n=1M-1|u(tk,xn)-unk|2h)1/2
of the numerical solutions for different values of M and N, where u(tk,xn) represents the exact solution and unk represents the numerical solution at (tk,xn). Table 1 gives the error analysis between the exact solution and solutions derived by difference schemes for N=M=20, 40, and 60, respectively.

Error analysis.

Difference schemes

N=M=20

N=M=40

N=M=60

Difference scheme (2.5)

0.0049

0.0025

0.0012

Difference scheme (2.6)

3.7155e-005

9.4107e-006

2.3679e-006

4. Conclusion

In this work, the first and second orders of accuracy difference schemes for the approximate solution of the Bitsadze-Samarskii nonlocal boundary value problem for elliptic equations are presented. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference schemes for elliptic equations are proved. The theoretical statements for the solution of these difference schemes are supported by the results of numerical examples. The second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme. As a future work, high orders of accuracy difference schemes for the approximate solutions of this problem could be established. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference schemes for elliptic equations could be proved.

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