# Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in Hölder Spaces

Authors: Okan Gercek
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A first order of accuracy difference scheme for the
approximate solution of abstract nonlocal boundary value problem−d2u(t)/dt2+sign(t)Au(t)=g(t), (0≤t≤1), du(t)/dt+sign(t)Au(t)=f(t), (−1≤t≤0), u(0+)=u(0−),u′(0+)=u′(0−),andu(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 without a weight is established. Moreover, as applications, coercivity estimates in Hölder norms
for the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.


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## Body

## 1.Introduction

Nonlocal boundary value problems for partial differential equations have been applied by various researchers in order to model numerous processes in different fields of applied sciences when they are unable to determine the boundary values of the unknown function (see, e.g., [1–15] and the references therein).Well-posedness of difference schemes of elliptic-parabolic equations with nonlocal boundary conditions in Hölder spaces with a weight was studied in [16–19].In paper [20], the well-posedness of abstract nonlocal boundary value problem
(1.1)-d2u(t)dt2+sign(t)Au(t)=g(t),(0≤t≤1),du(t)dt+sign(t)Au(t)=f(t),(-1≤t≤0),u(0+)=u(0-),u′(0+)=u′(0-),u(1)=u(-1)+μ
in Hölder spaces without a weight was established. The coercivity inequalities for solutions of the boundary value problem for elliptic-parabolic equations were obtained.In the present paper, the first order of accuracy difference scheme(1.2)-τ-2(uk+1-2uk+uk-1)+Auk=gk,gk=g(tk),tk=kτ,1≤k≤N-1,τ-1(uk-uk-1)-Auk-1=fk,fk=f(tk-1),tk-1=(k-1)τ,-N+1≤k≤-1,uN=u-N+μ,u1-u0=u0-u-1
for the approximate solution of problem (1.1) is considered. The well-posedness of difference scheme (1.2) in Hölder spaces without a weight is established. As an application, coercivity inequalities for solutions of difference scheme for elliptic-parabolic equations are obtained.Throughout the paper,H denotes a Hilbert space and A is a self-adjoint positive definite operator with A≥δI for some δ>δ0>0. Then, it is wellknown that B=(1/2)(τA+A(4+τ2A)) is a self-adjoint positive definite operator and B≥δ1/2I. Furthermore, R=(I+τB)-1 and P=P(τA)=(I+τA)-1 which are defined on the whole space H, are bounded operators, where I is the identity operator.


## 2.Well-Posedness of (1.2)

First of all, let us start with some auxiliary lemmas that are used throughout the paper.Lemma 2.1.
The following estimates are satisfied [19, 21, 22]:
(2.1)∥Pk∥H→H≤M(δ)(1+δτ)-k,kτ∥APk∥H→H≤M(δ),∥Rk∥H→H≤M(δ)(1+δτ)-k,kτ∥BRk∥H→H≤M(δ),∥Pk-e-kτA∥H→H≤M(δ)k,∥Rk-e-kτA1/2∥H→H≤M(δ)k,∥(I-R2N)-1∥H→H≤M(δ),k≥1,δ>0,
for some M(δ)>0, which is independent of τ is a positive small number.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, C([a,b]τ,H), Cα([-1,1]τ,H),  Cα/2([-1,0]τ,H), and Cα([0,1]τ,H)(0<α<1) denote Banach spaces on Fτ(H) with norms:
(2.2)∥φτ∥C([a,b]τ,H)=maxNa≤k≤Nb∥φk∥H,∥φτ∥Cα([-1,1]τ,H)=∥φτ∥C([-1,1]τ,H)+sup-N≤k<k+r≤0∥φk+r-φk∥Hr-α/2+sup1≤k<k+r≤N-1∥φk+r-φk∥Hr-α,∥φτ∥Cα/2([-1,0]τ,H)=∥φτ∥C([-1,0]τ,H)+sup-N≤k<k+r≤0∥φk+r-φk∥Hr-α/2,∥φτ∥Cα([0,1]τ,H)=∥φτ∥C([0,1]τ,H)+sup1≤k<k+r≤N-1∥φk+r-φk∥Hr-α.With the help of the self-adjoint positive definite operatorB in a Hilbert space H, the Banach space Eα=Eα(B,H)(0<α<1) consists of those v∈H for which the norm (see [22, 23]):
(2.3)∥v∥Eα=supz>0zα∥B(z+B)-1v∥H+∥v∥H,
is finite. By the definition of Eα(B,H),
(2.4)D(B)⊂Eα(B,H)⊂Eβ(B,H)⊂H,
for all β<α.Lemma 2.2.
For0<α<1, the norms of the spaces Eα(B,H) and Eα/2(A,H) are equivalent (see [24]).Theorem 2.3.
Supposeμ∈D(A),Aμ∈Eα(B,H),f0+g0∈Eα/2(A,H),f-N+gN∈Eα(B,H), g(t)∈Cα([0,1]τ,H), and f(t)∈Cα/2([-1,0]τ,H),0<α<1. Boundary value problem (1.2) is wellposed in Hölder space Cα([-1,1]τ,H) and the following coercivity inequality holds:
(2.5)∥{τ-2(uk+1-2uk+uk-1)}1N-1∥Cα([0,1]τ,H)+∥{Auk}-NN-1∥Cα([-1,1]τ,H)+∥{τ-1(uk-uk-1)}-N+10∥Cα/2([-1,0]τ,H)≤M[{1α(1-α)}∥Aμ∥Eα(B,H)+1α(1-α)[∥fτ∥Cα/2([-1,0]τ,H)+∥gτ∥Cα([0,1]τ,H)]+∥(I+τB)(f0+g0)∥Eα/2(A,H)+∥(I+τB)(f-N+gN)∥Eα(B,H){1α(1-α)}],
where M is independent of not only fτ, gτ, and   μ but also of τ and α.Proof.
First of all, let us get the formulae for solution of problem (1.2). By [21, 25],
(2.6)uk=(I-R2N)-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
is the solution of boundary value difference problem:
(2.7)-τ-2(uk+1-2uk+uk-1)+Auk=gkgk=g(tk),tk=kτ,1≤k≤N-1,u0=ξ,uN=ψ,(2.8)uk=P-kξ-τ∑s=k+10‍Ps-kfs,-N≤k≤-1
is the solution of inverse Cauchy problem:
(2.9)τ-1(uk-uk-1)-Auk-1=fk,fk=f(tk-1),tk-1=(k-1)τ,-(N-1)≤k≤0,u0=ξ.
Combining the conditionsψ=u-N+μ,ξ=u0 and formulas (2.6), (2.8), we get formulas
(2.10)uk=(I-R2N)-1{{∑}[Rk-R2N-k]u0+[RN-k-RN+k][PNu0-τ∑s=-N+10‍Ps+Nfs+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,(2.11)uk=P-ku0-τ∑s=k+10‍Ps-kfs,-N≤k≤-1.
Operator equation(2.12)2u0-Pu0+τPf0=(I-R2N)-1{{∑}[R-R2N-1]u0+[RN-1-RN+1](I-R2N)-1=×[PNu0-τ∑s=-N+10‍Ps+Nfs+μ](I-R2N)-1=-[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
follows from formulas (2.10), (2.11), and the condition u1-u0=u0-u-1. As the operator
(2.13)I+(I+τA)(I+2τA)-1R2N-1+B-1A(I+2τA)-1(I-R2N-1)-(2I+τB)(I+2τA)-1RNPN-1
has an inverse
(2.14)Tτ=(I+(I+τA)(I+2τA)-1R2N-1+B-1A(I+2τA)-1(I-R2N-1)-(2I+τB)(I+2τA)-1RNPN-1)-1,
it follows that
(2.15)u0=Tτ(I+2τA)-1(I+τA){{(2+τB)RN[-τ∑s=-N+10‍Ps+Nfs+μ]-RN-1B-1∑s=1N-1[RN-s-RN+s]gsτ{[-τ∑s=-N+10‍Ps+Nfs+μ]}}+(I-R2N)B-1∑s=1N-1Rs-1gsτ-(I-R2N)(I+τB)B-1Pf0{[∑]}}
for the solution of operator equation (2.12). Hence, we have formulas (2.10), (2.11), and (2.15) for the solution of difference problem (1.2).
Using formulae (2.10) and (2.15), we can get
(2.16)Au0=Tτ(I+2τA)-1(I+τA)×{{(2+τB)RN[-τ∑s=-N+10‍APs+N(fs-f-N+1)+Aμ]-RN-1AB-2{∑s=1N-1BRN-s(gs-gN-1)τ+∑s=1N-1{[∑]}BRN+s(g1-gs)τ}{∑}}+(I-R2N)AB-2∑s=1N-1BRs-1(gs-g1)τ{[∑]}}+Tτ(I+2τA)-1(I+τA)×{{(2+τB)RN(PN-I)f-N+1-RN-1AB-2{(I-RN-1)gN-1-(RN-2-R2N-1)g1}}+(I-R2N)AB-2(I-RN-1)g1-(I-R2N)(I+τB)B-1APf0},(2.17)AuN=PN{{∑}Tτ(I+2τA)-1(I+τA)×{{(2+τB)RN[-τ∑s=-N+10APs+N(fs-f-N+1)+Aμ]-RN-1AB-2{∑s=1N-1BRN-s(gs-gN-1)τ+∑s=1N-1BRN+s(g1-gs)τ}}+(I-R2N)AB-2∑s=1N-1BRs-1(gs-g1)τ{[∑s=-N+10]}}}-τ∑s=-N+10APs+N(fs-f-N+1)+Aμ+(PN-I)f-N+1+PN{Tτ(I+2τA)-1(I+τA)×{{(2+τB)RN(PN-I)f-N+1-RN-1AB-2{(I-RN-1)gN-1-(RN-2-R2N-1)g1}}+(I-R2N)AB-2(I-RN-1)g1-(I-R2N)(I+τB)B-1APf0}}.
Finally, we will get coercivity estimate (2.5). It is based on estimates
(2.18)∥{τ-2(uk+1-2uk+uk-1)}1N-1∥Cα([0,1]τ,H)+∥{Auk}1N-1∥Cα([0,1]τ,H)≤M[1α(1-α)∥gτ∥Cα([0,1]τ,H)+∥Au0-g0∥Eα(B,H)+∥AuN-gN∥Eα(B,H)]
for the solution of boundary value difference problem (2.7),
(2.19)∥{τ-1(uk-uk-1)}-N+10∥Cα/2([-1,0]τ,H)+∥{Auk}-N0∥Cα/2([-1,0]τ,H)≤M[1(α/2)(1-α/2)∥fτ∥Cα/2([-1,0]τ,H)+∥Au0+f0∥Eα(A,H)],
for the solution of inverse Cauchy difference problem (2.9), and
(2.20)∥Au0+f0∥Eα/2(A,H)≤Mα(1-α)[∥g∥Cα([0,1],H)+∥f∥Cα/2([-1,0],H)]+M[∥Aμ∥Eα(B,H)+∥f0+g0∥Eα/2(A,H)],∥Au0-g0∥Eα(B,H)≤Mα(1-α)[∥f∥Cα/2([-1,0],H)+∥g∥Cα([0,1],H)]+M[∥Aμ∥Eα(B,H)+∥f0+g0∥Eα/2(A,H)],∥AuN-gN∥Eα(B,H)≤Mα(1-α)[∥f∥Cα/2([-1,0],H)+∥g∥Cα([0,1],H)]+M[∥Aμ∥Eα(B,H)+∥f0+g0∥Eα/2(A,H)+∥f-N+gN∥Eα(B,H)]
for the solution of problem (1.2). Estimates (2.18) and (2.19) were established in [21, 25], respectively.
Estimates (2.20) are derived from the formulas (2.16) and (2.17) for the solution of problem (1.2), estimates (2.1) and following estimates
(2.21)∥Rk(τB)∥H→H≤M,1≤k≤N,∥(I-R2N(τB))-1∥H→H≤M,k≥1,∥Rk+r(τB)-Rk(τB)∥H→H≤M(r)α(k+r)α,1≤k<k+r≤N,0≤α≤1,∥(I-R(τB))2(τB)-2∥H→H≤M,∥(I+R(τB))-1∥H→H≤M,∥Tτ∥H→H≤M,∥BRPTτ∥H→H≤M,
which were established in [26]. This finalizes the proof of Theorem 2.3.


## 3.An Application

In this section, an application of  the abstract Theorem2.3 is considered. 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 mixed boundary value problem for multidimensional mixed equation:
(3.1)-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,f(0,x)+g(0,x)=0,f(-1,x)+g(1,x)=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. Here, ar(x)(x∈Ω),μ(x)(μ(x)=0,x∈S),g(t,x)(t∈(0,1),x∈Ω¯), andf(t,x)(t∈(-1,0),x∈Ω¯) are given smooth functions and ar(x)≥a>0.The discretization of problem (3.1) is carried out in two steps. In the first step, the grid sets
(3.2)Ω~h={x=xm=(h1m1,…,hnmn),m=(m1,…,mn),0≤mr≤Nr,hrNr=1,r=1,…,n},Ωh=Ω~h∩Ω,Sh=Ω~h∩Sare defined. To the differential operatorA generated by problem (3.1), the difference operator Ahx is assigned by formula:
(3.3)Ahxuh=-∑r=1n‍(ar(x)ux-hr)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
(3.4)-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∈Ω~h,uh(0+,x)=uh(0-,x),duh(0+,x)dt=duh(0-,x)dt,x∈Ω~h,
for an infinite system of ordinary differential equations.In the second step, problem (3.4) is replaced by difference scheme (1.2) (see [21]):
(3.5)-uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)=gkh(x),gkh(x)=gh(tk,x),tk=kτ,1≤k≤N-1,Nτ=1,x∈Ωh,ukh(x)-uk-1h(x)τ-Ahxuk-1h(x)=fkh(x),fkh(x)=fh(tk,x),tk-1=(k-1)τ,-N+1≤k≤-1,x∈Ωh,u-Nh(x)=uNh(x)+μh(x),x∈Ω~h,u1h(x)-u0h(x)=u0h(x)-u-1h(x),x∈Ω~h.
To formulate the result, we introduce the Hilbert spaces L2h=L2(Ω~h),W2h1=W21(Ω~h), and W2h2=W22(Ω~h) of the grid functions  φh(x)={φ(h1m1,…,hnmn)} defined on Ω~h, equipped with the norms:
(3.6)∥φh∥L2h=(∑x∈Ω~h‍|φh(x)|2h1⋯hn)1/2,∥φh∥W2h1=∥φh∥L2h+(∑x∈Ω~h‍∑r=1n‍|(φh)xr|2h1⋯hn)1/2,∥φh∥W2h2=∥φh∥L2h+(∑x∈Ω~h‍∑r=1n‍|(φh)xr|2h1⋯hn)1/2+(∑x∈Ω~h‍∑r=1n‍|(φh)xrx¯r,mr|2h1⋯hn)1/2.Theorem 3.1.
Letτ and |h|=h12+···+hn2 be sufficiently small numbers. Then, the solutions of difference scheme (3.5) satisfy the following coercivity stability estimate:
(3.7)∥{τ-2(uk+1h-2ukh+uk-1h)}1N-1∥Cα([0,1]τ,L2h)+∥{τ-1(ukh-uk-1h)}-N+10∥Cα/2([-1,0]τ,L2h)+∥{ukh}-NN-1∥Cα([-1,1]τ,W2h2)≤M[∥μh∥W2h2+1α(1-α)[∥{fkh}-N+1-1∥Cα/2([-1,0]τ,L2h)+∥{gkh}1N-1∥Cα([0,1]τ,L2h)]],
where M is not dependent on τ,h,μh(x),gkh(x),1≤k≤N-1, and fkh,-N+1≤k≤0.The proof of Theorem3.1 is based on Theorem 2.3, the symmetry properties of the difference operator Ahx defined by formula (3.3), and along with the following theorem on the coercivity inequality for the solution of elliptic difference equation in L2h.Theorem 3.2.
For the solution of elliptic difference problem:(3.8)Ahxuh(x)=ωh(x),x∈Ωh,uh(x)=0,x∈Sh,
the following coercivity inequality holds [27]:
(3.9)∑r=1n‍∥(uh)xrx-r,mr∥L2h≤M∥ωh∥L2h.
Here, M depends neither on h nor wh(x).



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