On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-Parabolic Equations

and Applied Analysis 3 on the whole space H, is a bounded operator. Here, I is the identity operator. The following operators D ( I τA τA 2 2 ) , G ( I − τ 2A 2 ) , P ( I τ 2 A ) , R I τB −1, Tτ ( I B−1A ( I τA τ 2 P−2 ) K ( I − R2N−1 ) GKP−2R2N−1 −GKP−2 2I τB R [ n ∑ i 1 αi ( I ( λi − [ λi τ ] τ ) A ) D − λi/τ u0 ])−1


Introduction
Methods of solutions of nonlocal boundary value problems for mixed-type differential equations have been studied extensively by various researchers see, e.g., 1-19 and the references therein .
In 20 , we considered the well-posedness of the following multipoint nonlocal boundary value problem: Abstract and Applied Analysis in a Hilbert space H with the self-adjoint positive definite operator A under assumption The well-posedness of multipoint nonlocal boundary value problem 1.1 in H ölder spaces with a weight was established.Moreover, coercivity estimates in H ölder norms for the solutions of nonlocal boundary value problems for elliptic-parabolic equations were obtained.
In 21 , we studied the well-posedness of the first order of accuracy difference scheme for the approximate solution of boundary value problem 1.1 under assumption 1.2 .
Throughout this work, we consider the following second order of accuracy difference scheme: for the approximate solution of boundary value problem 1.1 under assumption 1.2 .The well-posedness of difference scheme 1.3 in H ölder spaces with a weight is established.As an application, the stability, almost coercivity stability, and coercivity stability estimates for solutions of second order of accuracy difference scheme for the approximate solution of the nonlocal boundary elliptic-parabolic problem are obtained.

Main Theorems
Throughout the paper, H is a Hilbert space and we denote B 1/2 τA A 4 τ 2 A , where A is a self-adjoint positive definite operator.Then, it is clear that B is the self-adjoint positive definite operator and B ≥ δ 1/2 I where δ > δ 0 > 0, and R I τB −1 , which is defined Abstract and Applied Analysis 3 on the whole space H, is a bounded operator.Here, I is the identity operator.The following operators exist and are bounded for a self-adjoint positive operator A. Here, Furthermore, positive constants will be indicated by M which can differ in time.On the other hand M i α, β, . . . is used to focus on the fact that the constant depends only on α, β, . . .and the subindex i is used to indicate a different constant.
First of all, let us start with some auxiliary lemmas from 16, 22-24 that are essential below.
Lemma 2.1.For a self-adjoint positive operator A, the following estimates are satisfied:

2.3
From these estimates, it follows that

4
Abstract and Applied Analysis Lemma 2.2.For any g k , 1 ≤ k ≤ N − 1 and f k , −N 1 ≤ k ≤ 0, the solution of problem 1.3 exists, and the following formulas hold: Now, we study well-posedness of problem 1.3 .Let F τ H F a, b τ , H be the linear space of mesh functions and C α 0 −1, 0 τ , H , 0 < α < 1 Banach spaces with the following norms: for the solution of the following boundary value problem:

2.8
By 24 , we have for the solution of an inverse Cauchy difference problem:

2.10
Then, the proof of Theorem 2.3 is based on stability inequalities 2.7 and 2.9 and on the following estimates:

2.12
Proof.We have for the solution of boundary value problem 2.8 see 22 , and we get for the solution of inverse Cauchy difference problem 2.10 see 24 .Then, the proof of Theorem 2.4 is based on almost coercivity inequalities 2.13 and 2.14 and on the following estimates: for the solution of boundary value problem 1.3 .Proofs of these estimates follow the scheme of the papers 23, 24 and rely on both formula 2.5 and estimates 2.3 and 2.4 .Theorem 2.4 is proved.

8 Abstract and Applied Analysis
Proof.By 22, 24 , we have for the solution of boundary value problem 2.8 , and for the solution of inverse Cauchy difference problem 2.10 , respectively.Then, the proof of Theorem 2.5 is based on coercivity inequalities 2.17 -2.19 and the following estimates: for the solution of difference scheme 1.3 .Proofs of these estimates follow the scheme of the papers 22, 24 and rely on both estimates 2.3 and 2.4 and formula 2.5 .This concludes the proof of Theorem 2.5.

An Application
In this section, an application of these abstract Theorems 2.3, 2.4, and 2.5 is considered.
In −1, 1 × Ω, let us consider the following boundary value problem for multidimensional elliptic-parabolic equation: where a r x x ∈ Ω , ϕ x ϕ x 0, x ∈ S , g t, x t ∈ 0, 1 , x ∈ Ω , and f t, x t ∈ −1, 0 , x ∈ Ω are given smooth functions.Here, Ω is the unit open cube in the n-dimensional Euclidean space R n 0 < x k < 1, 1 ≤ k ≤ n with boundary S, Ω Ω ∪ S, and a r x a > 0. The discretization of problem 3.1 is carried out in two steps.In the first step, let us define the following grid sets:

3.2
We introduce the Hilbert spaces 2 Ω h of the grid functions ϕ h x {ϕ h 1 m 1 , . . ., h n m n } defined on Ω h , equipped with the following norms:

3.3
To the differential operator A generated by problem 3.1 , we assign the difference operator A x h by formula acting in the space of grid functions u h x , satisfying the conditions u h x 0 for all x ∈ S h .With the help of A x h , we arrive at the following nonlocal boundary value problem: for an infinite system of ordinary differential equations.
In the second step, we replace problem 3.5 by difference scheme 1.3 accurate to the following second order see 22, 24 :

3.6
Theorem 3.1.Let τ and |h| h 2 1 • • • h 2 n be sufficiently small positive numbers.Then, solutions of difference scheme 3.6 satisfy the following stability and almost coercivity estimates:

3.7
The proof of Theorem 3.1 is based on Theorem 2.3, Theorem 2.4, the symmetry property of the difference operator A x h defined by formula 3.4 , the estimate and the following theorem on the coercivity inequality for the solution of elliptic difference equation in L 2h .

Theorem 3.2. For the solution of the following elliptic difference problem:
A x h u h x ω h x , x ∈ Ω h , u h x 0, x ∈ S h , 3.9 the following coercivity inequality holds [25]: .

3.11
The proof of Theorem 3.3 is based on the abstract Theorem 2.5, Theorem 3.2, and the symmetry property of the difference operator A x h defined by formula 3.4 .

Theorem 3 . 3 .
Let τ and |h| be sufficiently small positive numbers.Then, solutions of difference scheme 3.6 satisfy the following coercivity stability estimates: of boundary value problem 1.3 .Estimates 2.11 follow from estimates 2.3 and 2.4 and formula 2.5 .This finishes the proof of Theorem 2.3.
Theorem 2.4.Assume that ϕ ∈ D A and f 0 , f −1 , g 1 ∈ D I τB .Then, for the solution of difference problem 1.3 , the following almost coercivity inequality holds: