# On Second Order of Accuracy Difference Scheme  of  the Approximate Solution of Nonlocal Elliptic-Parabolic Problems

Authors: Allaberen Ashyralyev; Okan Gercek
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

A second order of accuracy difference
scheme for the approximate solution of the abstract nonlocal boundary value
problem−d2u(t)/dt2+Au(t)=g(t), (0≤t≤1), du(t)/dt−Au(t)=f(t), (−1≤t≤0), u(1)=u(−1)+μ for differential equations in a Hilbert space H with a self-adjoint positive definite
operator A is considered. The well posedness of this difference scheme in Hölder
spaces is established. In applications, coercivity inequalities for the solution of a
difference scheme for elliptic-parabolic equations are obtained and a numerical
example is presented.


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## Body

## 1. Introduction

The role played by coercive inequalities in the study of boundary value problems for elliptic and parabolic partial differential equations is well known (see [1–4]).Nonlocal problems are widely used for mathematical modeling of various processes of physics, biology, chemistry, ecology, engineering, and industry when it is impossible to determine the boundary or initial values of the unknown function. Theory and numerical methods of solutions of the nonlocal boundary value problems for partial differential equations of variable type have been studied extensively by many researchers (see, e.g., [5–34] and the references therein).In paper [35], the nonlocal boundary value problem(1.1)-d2u(t)dt2+Au(t)=g(t)(0≤t≤1),du(t)dt-Au(t)=f(t)(-1≤t≤0),u(1)=u(-1)+μ
for the differential equation in a Hilbert space H with the self-adjoint positive definite operator A was considered. The well posedness of problem (1.1) in Hölder spaces was established. The first order of accuracy difference scheme for approximate solutions of nonlocal boundary value problem (1.1) was presented. In applications, the coercivity inequalities for solutions of difference schemes for elliptic-parabolic equations were obtained.In the present paper, the second order of accuracy difference scheme generated by Crank-Nicholson difference scheme(1.2)-τ-2(uk+1-2uk+uk-1)+Auk=gk,gk=g(tk),tk=kτ,1≤k≤N-1,Nτ=1,τ-1(uk-uk-1)-12(Auk-1+Auk)=fk,fk=f(tk-1/2),tk-1/2=(k-12)τ,-(N-1)≤k≤0,uN=u-N+μ,u2-4u1+3u0=-3u0+4u-1-u-2
for the approximate solution of problem (1.1) is presented. The well posedness of difference scheme (1.2) in Hölder spaces is established. As an application, coercivity inequalities for solutions of difference schemes for elliptic-parabolic equations are obtained. A numerical example is given.


## 2. The Formula for the Solution of Problem (1.2)

The following operators:(2.1)P=(I-τA2)G,G=(I+τA2)-1,R=(I+τB)-1,Tτ=((I-τ2A2)I+B-1A(I+τA+τ2G-2)K(I-R2N-1)+K(I-τ2A2)G-2R2N-1-K(I-τ2A2)G-2(2I+τB)RNPN)-1
exist and are bounded for a self-adjoint positive operator A. Here (2.2)B=12(τA+A(4+τ2A)),K=(I+2τA+54(τA)2)-1.Theorem 2.1.
For anygk,1≤k≤N-1, and fk,-N+1≤k≤0, the solution of problem (1.2) exists and the following formula holds:
(2.3)uk=(I-R2N)-1{∑s=-N+10[Rk-R2N-k]u0+[RN-k-RN+k][PNu0-τ∑s=-N+10Ps+N-1Gfs+μ]-[RN-k-RN+k](I+τB)(2I+τB)-1B-1∑s=1N-1[RN-s-RN+s]gsτ}+(I+τB)(2I+τB)-1B-1∑s=1N-1[R|k-s|-Rk+s]gsτ,1≤k≤N,(2.4)uk=P-ku0-τ∑s=k+10Ps-k-1Gfs,-N≤k≤-1,(2.5)u0=12TτKG-2×{(2I-τ2A){(2+τB)RN[-τ∑s=-N+10Ps+N-1Gfs+μ]-RN-1B-1∑s=1N-1[RN-s-RN+s]gsτ+(I-R2N)B-1∑s=1N-1Rs-1gsτ}+(I-R2N)(I+τB)(τB-1g1-4GB-1f0+PGB-1f0+GB-1f-1)∑s=1N-1},Tτ=((I-τ2A2)I+B-1A(I+τA+τ2G-2)K(I-R2N-1)+K(I-τ2A2)G-2R2N-1-K(I-τ2A2)G-2(2I+τB)RNPN)-1.Proof.
For any{fk}k=-N-1 and ξ, the solution of the auxiliary inverse Cauchy difference problem
(2.6)τ-1(uk-uk-1)-12(Auk-1+Auk)=fk,-(N-1)≤k≤0,u0=ξ
exists and the following formula holds [36]
(2.7)uk=P-kξ-τ∑s=k+10Ps-k-1Gfs,-N≤k≤-1.
Puttingξ=u0, we get (2.4).
Now, we consider the following auxiliary difference problem(2.8)-τ-2(uk+1-2uk+uk-1)+Auk=gk,gk=g(tk),tk=kτ,1≤k≤N-1,u0=ξ,uN=ψ.
It is well known that for the solution of  (2.8) the following formula holds [37, 38]:
(2.9)uk=(I-R2N)-1{∑s=1N-1[Rk-R2N-k]ξ+[RN-k-RN+k]ψ-[RN-k-RN+k](I+τB)(2I+τB)-1B-1∑s=1N-1[RN-s-RN+s]gsτ}+(I+τB)(2I+τB)-1B-1∑s=1N-1[R|k-s|-Rk+s]gsτ,1≤k≤N.
Applying  (2.7) and putting ξ=u0,ψ=PNu0-τ∑s=-N+10PN+s-1Gfs+μ, in (2.9), we get (2.3).
Foru0, using (2.3), (2.4), and the condition
(2.10)u2-4u1+3u0=-3u0+4u-1-u-2,
we obtain the operator equation
(2.11)(2I-τ2A){(I-R2N)-1{∑s=-N+10[R-R2N-1]u0+[RN-1-RN+1][PNu0-τ∑s=-N+10Ps+N-1Gfs+μ]-[RN-1-RN+1](I+τB)(2I+τB)-1B-1∑s=1N-1[RN-s-RN+s]gsτ}+(I+τB)(2I+τB)-1B-1∑s=1N-1[Rs-1-R1+s]gsτ}=-τ2g1+G2(2I+4τA+52(τA)2)u0+4Gτf0-PGτf0-Gτf-1.
The operator(2.12)I+B-1A(I+τA+τ2G-2)K(I-R2N-1)+K(I-τ2A2)G-2R2N-1-K(I-τ2A2)G-2(2I+τB)RNPN
has an inverse
(2.13)Tτ=((I-τ2A2)I+B-1A(I+τA+τ2G-2)K(I-R2N-1)+K(I-τ2A2)G-2R2N-1-K(I-τ2A2)G-2(2I+τB)RNPN)-1.
Hence, we obtain that(2.14)u0=12TτKG-2×{(2I-τ2A){(2+τB)RN[-τ∑s=-N+10Ps+N-1Gfs+μ]-RN-1B-1∑s=1N-1[RN-s-RN+s]gsτ+(I-R2N)B-1∑s=1N-1Rs-1gsτ}+(I-R2N)(I+τB)(τB-1g1-4GB-1f0+PGB-1f0+GB-1f-1)∑s=1N-1}.
This concludes the proof of Theorem2.1.


## 3. Main Theorems

Here, we study well posedness of problem (1.2). First, we give some necessary estimates for Pk,Rk, and Tτ. For a self-adjoint positive operator A, the following estimates are satisfied [36, 38, 39]:(3.1)∥Pk∥H→H≤1,∥G∥H→H≤1,kτ∥APkG2∥H→H≤M,k≥1,δ>0,(3.2)∥Rk∥H→H≤M(1+δτ)-k,kτ∥BRk∥H→H≤M,k≥1,δ>0,
where M is independent of τ.  From these estimates, it follows that(3.3)∥(I+B-1A(I+τA+τ2G-2)K(I-R2N-1)+K(I-τ2A2)G-2R2N-1-K(I-τ2A2)G-2(2I+τB)RNPN)-1)-1∥H→H≤M.LetFτ(H)=F([a,b]τ,H) be the linear space of mesh functions φτ={φk}NaNb defined on [a,b]τ={tk=kh,Na≤k≤Nb,Naτ=a,Nbτ=b} with values in the Hilbert space H. Next on Fτ(H) we denote C([a,b]τ,H)  and C0,1α([-1,1]τ,H),C0,1α([-1,0]τ,H),C0α([0,1]τ,H)(0<α<1) Banach spaces with the norms (3.4)∥φτ∥C([a,b]τ,H)=maxNa≤k≤Nb∥φk∥H,∥φτ∥C0,1α([-1,1]τ,H)=∥φτ∥C([-1,1]τ,H)+sup-N≤k<k+r≤0∥φk+r-φk∥E(-k)αrα+sup1≤k<k+r≤N-1∥φk+r-φk∥E((k+r)τ)α(N-k)αrα,∥φτ∥C0α([-1,0]τ,H)=∥φτ∥C([-1,0]τ,H)+sup-N≤k<k+r≤0∥φk+r-φk∥E(-k)αrα,∥φτ∥C0,1α([0,1]τ,H)=∥φτ∥C([0,1]τ,H)+sup1≤k<k+r≤N-1∥φk+r-φk∥E((k+r)τ)α(N-k)αrα.Nonlocal boundary value problem (1.2) is said to be stable in F([-1,1]τ,H) if we have the inequality (3.5)∥uτ∥F([-1,1]τ,H)≤M[∥fτ∥F([-1,0]τ,H)+∥gτ∥F([0,1]τ,H)+∥μ∥H],
where M is independent of not only fτ,gτ,andμ but also τ.Theorem 3.1.
Nonlocal boundary value problem (1.2) is stable in C([-1,1]τ,H) norm.Proof.
By [38], we have
(3.6)∥{uk}1N-1∥C([0,1]τ,H)≤M[∥gτ∥C([0,1]τ,H)+∥u0∥H+∥uN∥H]
for the solution of  boundary value problem (2.8).
By [36], we get
(3.7)∥{uk}-N0∥C([-1,0]τ,H)≤M[∥fτ∥C([-1,0]τ,H)+∥u0∥H]
for the solution of an inverse Cauchy  difference problem (2.6). Then, the proof of Theorem 3.1 is based on the stability inequalities (3.6), (3.7), and on the estimates
(3.8)∥u0∥H≤M[∥fτ∥C([-1,0]τ,H)+∥gτ∥C([1,0]τ,H)+∥μ∥H],(3.9)∥uN∥H≤M[∥fτ∥C([-1,0]τ,H)+∥gτ∥C([1,0]τ,H)+∥μ∥H],
for the solution of the boundary value problem (1.2). Estimates (3.8) and (3.9) follow from formula (2.5) and estimates (3.1), (3.2), and (3.3) which conclude the proof of Theorem 3.1.Theorem 3.2.
Assume thatμ∈D(A) and f0,f-1,g1∈D(I+τB). Then, for the solution of difference problem (1.2), we have the following almost coercivity inequality:
(3.10)∥{τ-2(uk+1-2uk+uk-1)}1N-1∥C([0,1]τ,H)+∥{τ-1(uk-uk-1)}-N+10∥C([-1,0]τ,H)+∥{Auk}1N-1∥C([0,1]τ,H)+∥{12(Auk+Auk-1)}-N+10∥C([-1,0]τ,H)≤M[min{ln1τ,1+|ln∥A∥H→H|}[∥fτ∥C([-1,0]τ,H)+∥gτ∥C([0,1]τ,H)]+∥Aμ∥H+∥(I+τB)f0∥H+∥(I+τB)g1∥H+∥(I+τB)f-1∥H1τ],
where M does not dependent on not only fτ,gτ,andμ but also τ.Proof.
By [40], we have
(3.11)∥{τ-1(uk-uk-1)}-N+10∥C([-1,0]τ,H)+∥{12(Auk+Auk-1)}-N+10∥C([-1,0]τ,H)≤M[min{ln1τ,1+|ln∥A∥H→H|}∥fτ∥C([-1,0]τ)+∥Au0∥H]
for the solution of an inverse Cauchy difference problem (2.6).
By [38], we get
(3.12)∥{τ-2(uk+1-2uk+uk-1)}1N-1∥C([0,1]τ,H)+∥{Auk}1N-1∥C([0,1]τ,H)≤M[min{ln1τ,1+|ln∥A∥H→H|}∥gτ∥C([0,1]τ,H)+∥Au0∥H+∥AuN∥H]
for the solution of boundary value problem (2.8).
Then, the proof of Theorem3.2 is based on almost coercivity inequalities (3.11), (3.12), and on the estimates
(3.13)∥Au0∥H≤M[{ln1τ,1+|ln∥A∥H→H|}∥Aμ∥H+∥(I+τB)f0∥H+min{ln1τ,1+|ln∥A∥H→H|}[∥fτ∥C([-1,0]τ,H)+∥gτ∥C([0,1]τ,H)]],∥AuN∥H≤M[{ln1τ,1+|ln∥A∥H→H|}[∥Aμ∥H+∥(I+τB)f0∥H]+min{ln1τ,1+|ln∥A∥H→H|}[∥fτ∥C([-1,0]τ,H)+∥gτ∥C([0,1]τ,H)]]
for the solution of boundary value problem (1.2). The proof of these estimates follows the scheme of the papers [38, 40] and relies on both the formula (2.5) and the estimates (3.1), (3.2), and (3.3).
This concludes the proof of Theorem3.2.LetC̃0,1α([-1,1]τ,H),C̃0α([-1,0]τ,H),0<α<1 be the Banach spaces with the norms (3.14)∥φτ∥C̃0,1α([-1,1]τ,H)=∥φτ∥C([-1,1]τ,H)+sup-N≤k<k+2r≤0∥φk+2r-φk∥E(-k)α(2r)α+sup1≤k<k+r≤N-1∥φk+2r-φk∥E((k+r)τ)α(N-k)αrα,∥φτ∥C̃0α([-1,0]τ,H)=∥φτ∥C([-1,0]τ,H)+sup-N≤k<k+2r≤0∥φk+2r-φk∥E(-k)α(2r)α.Theorem 3.3.
Let the assumptions of Theorem3.2 be satisfied. Then, boundary value problem (1.2) is well posed in Hölder spaces C0,1α([-1,1]τ,H) and C̃0,1α([-1,1]τ,H), and the following coercivity inequalities hold:
(3.15)∥{τ-2(uk+1-2uk+uk-1)}1N-1∥C0,1α([0,1]τ,H)+∥{τ-1(uk-uk-1)}-N+10∥C̃0α([-1,0]τ,H)+∥{Auk}1N-1∥C0,1α([0,1]τ,H)+∥{12(Auk+Auk-1)}-N+10∥C̃0α([-1,0]τ,H)≤M[1α(1-α)[∥fτ∥C0α([-1,0]τ,H)+∥gτ∥C0,1α([0,1]τ,H)]+∥Aμ∥H+∥(I+τB)f0∥H+∥(I+τB)g1∥H+∥(I+τB)f-1∥H1α(1-α)],∥{τ-2(uk+1-2uk+uk-1)}1N-1∥C0,1α([0,1]τ,H)+∥{τ-1(uk-uk-1)}-N+10∥C̃0α([-1,0]τ,H)+∥{Auk}1N-1∥C0,1α([0,1]τ,H)+∥{12(Auk+Auk-1)}-N+10∥C̃0α([-1,0]τ,H)≤M[1α(1-α)[∥fτ∥C̃0α([-1,0]τ,H)+∥gτ∥C0,1α([0,1]τ,H)]+∥Aμ∥H+∥(I+τB)f0∥H+∥(I+τB)g1∥H+∥(I+τB)f-1∥H1α(1-α)],
where M is independent of not only fτ,gτ, and μ but also τ and α.Proof.
By [39, 40],
(3.16)∥{τ-1(uk-uk-1)}-N+10∥C̃0α([-1,0]τ,H)+∥{12(Auk+Auk-1)}-N+10∥C̃0α([-1,0]τ,H)≤M[1α(1-α)∥fτ∥C0α([-1,0]τ,H)+∥Au0∥H],(3.17)∥{τ-1(uk-uk-1)}-N+10∥C̃0α([-1,0]τ,H)+∥{12(Auk+Auk-1)}-N+10∥C̃0α([-1,0]τ,H)≤M[1α(1-α)∥fτ∥C̃0α([-1,0]τ,H)+∥Au0∥H]
for the solution of an inverse Cauchy  difference problem (2.6) can be written.
By [37, 38], we get
(3.18)∥{τ-2(uk+1-2uk+uk-1)}1N-1∥C0,1α([0,1]τ,H)+∥{Auk}1N-1∥C0,1α([0,1]τ,H)≤M[1α(1-α)∥gτ∥C0,1α([0,1]τ,H)+∥Au0∥H+∥AuN∥H]
for the solution of boundary value problem (2.8).
Then, the proof of Theorem3.3 is based on coercivity inequalities (3.16)–(3.18), and the estimates
(3.19)∥Au0∥H≤M[1α(1-α)[∥fτ∥C̃0α([-1,0]τ,H)+∥gτ∥C0,1α([0,1]τ,H)]+∥Au∥H+∥(I+τB)f0∥H+∥(I+τB)g1∥H+∥(I+τB)f-1∥H1α(1-α)],(3.20)∥AuN∥H≤M[1α(1-α)[∥fτ∥C̃0α([-1,0]τ,H)+∥gτ∥C0,1α([0,1]τ,H)]+∥Aμ∥H+∥(I+τB)f0∥H+∥(I+τB)g1∥H+∥(I+τB)f-1∥H1α(1-α)]
for the solution of boundary value problem (1.2).
Estimates (3.19) and (3.20) follow from the formulas
(3.21)Au0=12TτKG-2×{(2I-τ2A){(2+τB)RN[-τ∑s=-N+10APs+N-1G(fs-f-N+1)+Aμ]-RN-1AB-1∑s=1N-1RN-s(gs-gN-1)τ+RN-1AB-1∑s=1N-1RN+s(gs-g1)τ+(I-R2N)AB-1∑s=1N-1Rs-1(gs-g1)τ}+(I-R2N)(I+τB)(τB-1Ag1-4GB-1Af0+PGB-1Af0+GB-1Af-1)+(2I-τ2A)(2+τB)RN(PN-I)f-N+1+AB-2(RN-1-I){RN-1gN-1+(R2N-R2N-1-I)g1}[-τ∑s=-N+10APs+N-1G(fs-f-N+1)+Aμ]},AuN=12PNTτKG-2×{(2I-τ2A){(2+τB)RN[-τ∑s=-N+10APs+N-1G(fs-f-N+1)+Aμ]-RN-1AB-1∑s=1N-1RN-s(gs-gN-1)τ+RN-1AB-1∑s=1N-1RN+s(gs-g1)τ+(I-R2N)AB-1∑s=1N-1BRs-1(gs-g1)τ}+(I-R2N)(I+τB)(τB-1Ag1-4GB-1Af0+PGB-1Af0+GB-1Af-1)+(2I-τ2A)(2+τB)RN(PN-I)f-N+1+AB-2(RN-1-I){RN-1gN-1+(R2N-R2N-1-I)g1}[-τ∑s=-N+10APs+N-1G(fs-f-N+1)+Aμ]}-τ∑s=-N+10APs+N-1G(fs-f-N+1)+Aμ+(PN-I)f-N+1
for the solution of problem (1.2)  and estimates (3.1), (3.2), and (3.3).
This finalizes the proof of Theorem3.3.


## 4. Applications

In this section, we indicate applications of Theorems3.1, 3.2, and 3.3 to obtain the stability, the almost coercive stability, and the coercive stability estimates for the solutions of these difference schemes for the approximate solution of nonlocal mixed problems. First, let Ω be the unit open cube in the n-dimensional Euclidean space ℝn(0<xk<1,1≤k≤n) with boundary S,Ω¯=Ω∪S. In [-1,1]×Ω, the boundary value problem for the multidimensional elliptic-parabolic equation(4.1)-utt-∑r=1n(ar(x)uxr)xr=g(t,x),0<t<1,x∈Ω,ut+∑r=1n(ar(x)uxr)xr=f(t,x),-1<t<0,x∈Ω,u(t,x)=0,x∈S,-1≤t≤1,u(1,x)=u(-1,x)+μ(x),x∈Ω¯,u(0+,x)=u(0-,x),ut(0+,x)=ut(0-,x),x∈Ω¯
is considered. Problem (4.1) has a unique smooth solution u(t,x) for f(t,x)(t∈(-1,0),x∈Ω¯),g(t,x)(t∈(0,1),x∈Ω¯) the smooth functions, and ar(x)⩾a>0(x∈Ω).The discretization of problem (4.1) is carried out in two steps. In the first step, the grid sets(4.2)Ω̃h={x=xm=(h1m1,…,hnmn),m=(m1,…,mn),0≤mr≤Nr,hrNr=1,r=1,…,n},Ωh=Ω̃h∩Ω,Sh=Ω̃h∩S
are defined. To the differential operator A generated by problem (4.1), we assign the difference operator Ahx by the formula(4.3)Ahxuxh=-∑r=1n(ar(x)ux̅rh)xr,mr
acting in the space of grid functions uh(x), satisfying the conditions uh(x)=0 for all x∈Sh. With the help of Ahx, we arrive at the nonlocal boundary value problem(4.4)-d2uh(t,x)dt2+Ahxuh(t,x)=gh(t,x),0<t<1,x∈Ωh,duh(t,x)dt-Ahxuh(t,x)=fh(t,x),-1<t<0,x∈Ωh,uh(1,x)=uh(-1,x)+μh(x),x∈Ω̃h,uh(0+,x)=uh(0-,x),duh(0+,x)dt=duh(0-,x)dt,x∈Ω̃h
for an infinite system of ordinary differential equations.Replacing problem (4.4) by the difference scheme (1.2), one can obtain the second order of accuracy difference scheme(4.5)-uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)=gkh(x),gkh(x)=g(tk,xn),tk=kτ,1≤k≤N-1,Nτ=1,x∈Ωh,ukh(x)-uk-1h(x)τ-Ahx2(ukh(x)+uk-1h(x))=fkh(x),fkh(x)=f(tk-1/2,xn),tk-1/2=(k-12)τ,-N+1≤k≤0,x∈Ωh,uNh(x)=u-Nh(x)+μh(x),x∈Ω̃h,-u2h(x)+4u1h(x)-3u0h(x)=3u0h(x)-4u-1h(x)+u-2h(x),x∈Ω̃h.Let us give a corollary of Theorems3.1 and 3.2.Theorem 4.1.
Letτ and |h|=h12+···+hn2 be sufficiently small positive numbers. Then, solutions of difference scheme (4.5) satisfy the following stability and almost coercivity estimates:
(4.6)∥{ukh}-NN-1∥C([-1,1]τ,L2h)≤M[∥{fkh}-N+1-1∥C([-1,0]τ,L2h)+∥{gkh}1N-1∥C([0,1]τ,L2h)+∥μh∥L2h],∥{τ-2(uk+1h-2ukh+uk-1h)}1N-1∥C([0,1]τ,L2h)+∥{ukh}1N-1∥C([0,1]τ,W2h2)+∥{τ-1(ukh-uk-1h)}-N+10∥C([-1,0]τ,L2h)+∥{ukh+uk-1h2}-N+10∥C([-1,0]τ,W2h2)≤M[ln1τ+|h|[∥{fkh}-N+1-1∥C([-1,0]τ,L2h)+∥{gkh}1N-1∥C([0,1]τ,L2h)]∥f0h∥L2h+∥f-1h∥L2h+∥g1h∥L2h+∥μh∥W2h2+τ∥f0h∥W2h1+τ∥f-1h∥W2h1+τ∥g1h∥W2h1+ln1τ+|h|[∥{fkh}-N+1-1∥C([-1,0]τ,L2h)+∥{gkh}1N-1∥C([0,1]τ,L2h)]].
Here, M is independent of not only τ,h,μh(x) but also gkh(x),1≤k≤N-1, and fkh,-N+1≤k≤0.The proof of Theorem4.1 is based on Theorems 3.1 and 3.2, the estimate(4.7)min{ln1τ,1+|ln∥Ahx∥L2h→L2h|}≤Mln1τ+|h|,
the symmetry properties of the difference operator Ahx defined by formula (4.3) in L2h, and the following theorem.Theorem 4.2.
For the solution of the elliptic difference problem(4.8)Ahxuh(x)=ωh(x),x∈Ωh,uh(x)=0,x∈Sh
the following coercivity inequality holds [41]:
(4.9)∑r=1n∥(uh)x̅rxr,mr∥L2h≤M∥ωh∥L2h.Let us give a corollary of Theorem3.3.Theorem 4.3.
Letτ and |h| be sufficiently small positive numbers. Then, solutions of difference scheme (4.5) satisfy the following coercivity stability estimates:
(4.10)∥{τ-2(uk+1h-2ukh+uk-1h)}1N-1∥C0,1α([0,1]τ,L2h)+∥{τ-1(ukh-uk-1h)}-N+10∥C̃0α([-1,0]τ,L2h)+∥{ukh}1N-1∥C0,1α([0,1]τ,W2h2)+∥{ukh+uk-1h2}-N+10∥C̃0α([-1,0]τ,W2h2)≤M[[∥{fkh}-N+1-1∥C0α([-1,0]τ,L2h)+∥{gkh}1N-1∥C0,1α([0,1]τ,L2h)]∥μh∥W2h2+τ∥f0h∥W2h1+τ∥f-1h∥W2h1+τ∥g1h∥W2h1+1α(1-α)[∥{fkh}-N+1-1∥C0α([-1,0]τ,L2h)+∥{gkh}1N-1∥C0,1α([0,1]τ,L2h)]],∥{τ-2(uk+1h-2ukh+uk-1h)}1N-1∥C0,1α([0,1]τ,L2h)+∥{ukh+uk-1h2}-N+10∥C̃0α([-1,0]τ,W2h2)+∥{τ-1(ukh-uk-1h)}-N+10∥C̃0α([-1,0]τ,L2h)+∥{ukh}1N-1∥C0,1α([0,1]τ,W2h2)≤M[[∥{fkh}-N+1-1∥C0α([-1,0]τ,L2h)+∥{gkh}1N-1∥C0,1α([0,1]τ,L2h)]∥μh∥W2h2+τ∥f0h∥W2h1+τ∥f-1h∥W2h1+τ∥g1h∥W2h1+1α(1-α)[∥{fkh}-N+1-1∥C̃0α([-1,0]τ,L2h)+∥{gkh}1N-1∥C0,1α([0,1]τ,L2h)]],
where M is independent of not only τ,h,andμh(x) but also gkh(x),1≤k≤N-1 andfkh,-N+1≤k≤0.The proof of Theorem4.3 is based on the abstract Theorems 3.3 and 4.2, and the symmetry properties of the difference operator Ahx defined by the formula (4.3).Second, the mixed boundary value problem for the elliptic-parabolic equation(4.11)-utt-(a(x)ux)x+δu=g(t,x),0<t<1,0<x<1,ut+(a(x)ux)x-δu=f(t,x),-1<t<0,0<x<1,u(t,0)=u(t,1),ux(t,0)=ux(t,1),-1≤t≤1,u(1,x)=u(-1,x)+μ(x),0≤x≤1,u(0+,x)=u(0-,x),ut(0+,x)=ut(0-,x),0≤x≤1
is considered. Problem (4.11) has a unique smooth solution u(t,x) for f(t,x)(t∈[-1,0],x∈[0,1]),g(t,x)(t∈[0,1],x∈[0,1]), the smooth functions, and a(x)⩾a>0(x∈(0,1)),δ=const>0.Note that in a similar manner one can construct the difference schemes of the second order of accuracy with respect to one variable for approximate solutions of the boundary value problem (4.11). Abstract theorems given above permit us to obtain the stability, the almost stability and the coercive stability estimates for the solutions of these difference schemes.


## 5. Numerical Results

We consider the nonlocal boundary value problem(5.1)∂u∂t+∂2u∂x2=sinx,-1<t≤0,0<x<π,∂2u∂t2+∂2u∂x2=sinx,0<t<1,0<x<π,u(1,x)=u(-1,x)+2sinh1sinx,0≤x≤π,u(t,0)=u(t,π)=0,-1≤t≤1
for the elliptic-parabolic equation.The exact solution of this problem isu(t,x)=(et-1)sinx.Now, we give the results of the numerical analysis. The errors computed by(5.2)EMN=max-N≤k≤N,1≤n≤M-1|u(tk,xn)-unk|
of the numerical solutions are given inTable 1.Table 1
Comparison of the errors.
MethodN=M=20N=M=30N=M=601st order of accuracy d. s.0.0435410.0305150.0159732nd order of accuracy d. s0.0006270.0002830.000071Thus, the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme.



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*Source: 705172-2010-07-26.xml*