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

Abstract

We are interested in studying a second order of accuracy implicit difference scheme for the solution of the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of this difference scheme is established. In an application, coercivity estimates in Hölder norms for approximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolic differential equations are obtained.

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: −𝑑²𝑢(𝑡)𝑑𝑡²(𝑡)(𝑡),(),(𝑡)+𝐴𝑢=𝑔0≤𝑡≤1𝑑𝑢(𝑡)(𝑡),(),𝑢(1)=𝑑𝑡−𝐴𝑢=𝑓−1≤𝑡≤0_𝐽𝑖=1𝛼_𝑖𝑢𝜆_𝑖+𝜑,−1≤𝜆₁<𝜆₂<⋯<𝜆_𝑖<⋯<𝜆_𝐽≤0, (1.1) in a Hilbert space 𝐻 with the self-adjoint positive definite operator 𝐴 under assumption ^𝐽_𝑖=1𝛼_𝑖≤1. (1.2)

The well-posedness of multipoint nonlocal boundary value problem (1.1) in Hölder spaces with a weight was established. Moreover, coercivity estimates in Hölder 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: −𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1+𝐴𝑢_𝑘=𝑔_𝑘,𝑔_𝑘=𝑔𝑡_𝑘,𝑡_𝑘𝜏=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,₋₁𝑢_𝑘−𝑢_𝑘−1−𝜏𝐼+2𝐴𝐴𝑢_𝑘−1=𝜏𝐼+2𝐴𝑓_𝑘,𝑓_𝑘=𝑓𝑡_𝑘−1/2,𝑡_𝑘−1/2=1𝑘−2()𝑢𝜏,−𝑁−1≤𝑘≤0,₂−4𝑢₁+3𝑢₀=−3𝑢₀+4𝑢₋₁−𝑢₋₂,𝑢_𝑁=^𝐽_𝑘=1𝛼_𝑖𝑢_{[𝜆𝑖/𝜏]}+𝜆_𝑖−𝜆_𝑖𝜏𝜏𝑓_{[𝜆𝑖/𝜏]}+𝐴𝑢_{[𝜆𝑖/𝜏]}+𝜑, (1.3) for the approximate solution of boundary value problem (1.1) under assumption (1.2).

The well-posedness of difference scheme (1.3) in Hölder 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.

2. Main Theorems

Throughout the paper, 𝐻 is a Hilbert space and we denote 𝐵=(1/2)(𝜏𝐴+𝐴(4+𝜏²𝐴)), 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^1/2𝐼 where 𝛿>𝛿₀>0, and 𝑅=(𝐼+𝜏𝐵)⁻¹, which is defined on the whole space 𝐻, is a bounded operator. Here, 𝐼 is the identity operator. The following operators ()𝐷=𝐼+𝜏𝐴+𝜏𝐴²2𝜏,𝐺=𝐼−²𝐴2𝜏,𝑃=𝐼+2𝐴,𝑅=(𝐼+𝜏𝐵)⁻¹,𝑇_𝜏=𝐼+𝐵⁻¹𝐴𝜏𝐼+𝜏𝐴+2𝑃⁻²𝐾𝐼−𝑅^2𝑁−1+𝐺𝐾𝑃⁻²𝑅^2𝑁−1−𝐺𝐾𝑃⁻²(2𝐼+𝜏𝐵)𝑅^𝑁𝑛_𝑖=1𝛼_𝑖𝐼+𝜆_𝑖−𝜆_𝑖𝜏𝜏𝐴𝐷^{𝑖−[𝜆/𝜏]}𝑢₀⁻¹ (2.1) exist and are bounded for a self-adjoint positive operator 𝐴. Here, 1𝐵=2𝜏𝐴+𝐴4+𝜏²𝐴5,𝐾=𝐼+2𝜏𝐴+4(𝜏𝐴)²⁻¹. (2.2)

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: 𝑅^𝑘_𝐻→𝐻≤𝑀₁(𝛿)(1+𝛿𝜏)^−𝑘,𝐷^𝑘_𝐻→𝐻≤𝑀₁(𝛿),𝐵𝑅^𝑘_𝐻→𝐻≤𝑀₁(𝛿),𝑃𝑘𝜏^{−1𝐻→𝐻}≤𝑀₁(𝛿),𝐴𝐷^𝑘_𝐻→𝐻≤𝑀₁(𝛿),𝐷𝑘𝜏_𝑘−𝑒^{−𝑘𝜏𝐴}_𝐻→𝐻≤𝑀₁(𝛿)𝑘²,𝐼−𝑅^2𝑁−1_𝐻→𝐻≤𝑀₁(𝛿),𝑅^𝑘−𝑒^{−𝑘𝜏𝐴1/2}_𝐻→𝐻≤𝑀₁(𝛿)𝑘,𝑘≥1,𝛿>0. (2.3) From these estimates, it follows that 𝐼+𝐵⁻¹𝐴𝜏𝐼+𝜏𝐴+2𝑃⁻²𝐾𝐼−𝑅^2𝑁−1+𝐺𝐾𝑃⁻²𝑅^2𝑁−1−𝐺𝐾𝑃⁻²(2𝐼+𝜏𝐵)×𝑅^𝑁𝑛_𝑖=1𝛼_𝑖𝐼+𝜆_𝑖−𝜆_𝑖𝜏𝜏𝐴𝐷^{−[𝜆𝑖/𝜏]−1}_𝐻→𝐻≤𝑀₂(𝛿). (2.4)

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: 𝑢_𝑘=𝐼−𝑅^2𝑁−1×𝑅^𝑘−𝑅^2𝑁−𝑘𝑢₀+𝑅^𝑁−𝑘−𝑅^𝑁+𝑘×^𝑛_𝑖=1𝛼_𝑖𝐼+𝜆_𝑖−𝜆_𝑖𝜏𝜏𝐴𝐷^{𝑖−[𝜆/𝜏]}𝑢₀−𝜏⁰_{𝑠=[𝜆𝑖/𝜏]+1}𝑃𝐷^{𝑠−[𝜆𝑖/𝜏]}𝑓_𝑠+𝜆_𝑖−𝜆_𝑖𝜏𝜏𝑓_{[𝜆𝑖/𝜏]}−𝑅+𝜑_𝑁−𝑘−𝑅^𝑁+𝑘(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑠=1𝑅^𝑁−𝑠−𝑅^𝑁+𝑠𝑔_𝑠𝜏+(𝐼+𝜏𝐵)(2𝐼+𝜏𝐵)⁻¹𝐵^−1𝑁−1_𝑠=1𝑅^{|𝑘−𝑠|}−𝑅^𝑘+𝑠𝑔_𝑠𝑢𝜏,1≤𝑘≤𝑁,_𝑘=𝐷^−𝑘𝑢₀−𝜏⁰_𝑠=𝑘+1𝑃𝐷^𝑠−𝑘𝑓_𝑠𝑢,−𝑁≤𝑘≤−1,₀=12𝑇_𝜏𝐾𝑃⁻²×2𝐼−𝜏²𝐴×(2+𝜏𝐵)𝑅^𝑁×^𝑛_𝑖=1𝛼_𝑖𝐼+𝜆_𝑖−𝜆_𝑖𝜏𝜏𝐴×𝐷^{𝑖−[𝜆/𝜏]}𝑢₀−𝜏⁰_{𝑠=[𝜆𝑖/𝜏]+1}𝑃𝐷^{𝑠−[𝜆𝑖/𝜏]}𝑓_𝑠+𝜆_𝑖−𝜆_𝑖𝜏𝜏𝑓_{[𝜆𝑖/𝜏]}+𝜑−𝑅^𝑁−1𝐵∑^−1𝑁−1_𝑠=1𝑅^𝑁−𝑠−𝑅^𝑁+𝑠𝑔_𝑠𝜏+𝐼−𝑅^2𝑁𝐵∑^−1𝑁−1_𝑠=1𝑅^𝑠−1𝑔_𝑠𝜏+𝐼−𝑅^2𝑁(𝐼+𝜏𝐵)𝜏𝐵⁻¹𝑔₁−4𝑃𝐵⁻¹𝑓₀+𝑃𝐷𝐵⁻¹𝑓₀+𝑃𝐵⁻¹𝑓₋₁,𝑇_𝜏=𝐼+𝐵⁻¹𝐴𝜏𝐼+𝜏𝐴+2𝑃⁻²𝐾𝐼−𝑅^2𝑁−1+𝐺𝐾𝑃⁻²𝑅^2𝑁−1−𝐺𝐾𝑃⁻²(2𝐼+𝜏𝐵)𝑅^𝑁𝑛_𝑖=1𝛼_𝑖𝐼+𝜆_𝑖−𝜆_𝑖𝜏𝜏𝐴𝐷^{𝑖−[𝜆/𝜏]}𝑢₀⁻¹. (2.5)

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 𝜑^𝜏={𝜑_𝑘}𝑁_𝑁 defined on [𝑎,𝑏]_𝜏={𝑡_𝑘=𝑘ℎ,  𝑁≤𝑘≤𝑁,  𝑁𝜏=𝑎,  𝑁𝜏=𝑏} with values in the Hilbert space 𝐻. Next, on 𝐹_𝜏(𝐻) we denote 𝐶([𝑎,𝑏]_𝜏,𝐻),𝐶^𝛼_0,1([−1,1]_𝜏,𝐻),𝐶^𝛼_0,1([−1,0]_𝜏,𝐻),  𝐶^𝛼₀([0,1]_𝜏,𝐻),  𝐶^𝛼_0,1([−1,1]_𝜏,𝐻), and 𝐶^𝛼₀([−1,0]_𝜏,𝐻),0<𝛼<1 Banach spaces with the following norms: 𝜑^𝜏_{𝐶([𝑎,𝑏]𝜏,𝐻)}=max_{𝑁𝑎≤𝑘≤𝑁𝑏}𝜑_𝑘𝐻,𝜑^𝜏_{𝐶𝛼0,1([−1,1]𝜏,𝐻)}=𝜑^𝜏_{𝐶([−1,1]𝜏,𝐻)}+sup_{−𝑁≤𝑘<𝑘+𝑟≤0}𝜑_𝑘+𝑟−𝜑_𝑘𝐸(−𝑘)^𝛼𝑟^−𝛼+sup_{1≤𝑘<𝑘+𝑟≤𝑁−1}𝜑_𝑘+𝑟−𝜑_𝑘𝐸((𝑘+𝑟)𝜏)^𝛼(𝑁−𝑘)^𝛼𝑟^−𝛼,𝜑^𝜏_{𝐶𝛼0([−1,0]𝜏,𝐻)}=𝜑^𝜏_{𝐶([−1,0]𝜏,𝐻)}+sup_{−𝑁≤𝑘<𝑘+𝑟≤0}𝜑_𝑘+𝑟−𝜑_𝑘𝐸(−𝑘)^𝛼𝑟^−𝛼,𝜑^𝜏_{𝐶𝛼0,1([0,1]𝜏,𝐻)}=𝜑^𝜏_{𝐶([0,1]𝜏,𝐻)}+sup_{1≤𝑘<𝑘+𝑟≤𝑁−1}‖𝜑_𝑘+𝑟−𝜑_𝑘‖_𝐸((𝑘+𝑟)𝜏)^𝛼(𝑁−𝑘)^𝛼𝑟^−𝛼,𝜑^𝜏𝐶^𝛼_{0,1([−1,1]𝜏,𝐻)}=𝜑^𝜏_{𝐶([−1,1]𝜏,𝐻)}+sup_{−𝑁≤𝑘<𝑘+2𝑟≤0}𝜑_𝑘+2𝑟−𝜑_𝑘𝐸(−𝑘)^𝛼(2𝑟)^−𝛼+sup_{1≤𝑘<𝑘+𝑟≤𝑁−1}𝜑_𝑘+𝑟−𝜑_𝑘𝐸((𝑘+𝑟)𝜏)^𝛼(𝑁−𝑘)^𝛼𝑟^−𝛼,𝜑^𝜏𝐶^𝛼_{0([−1,0]𝜏,𝐻)}=𝜑^𝜏_{𝐶([−1,0]𝜏,𝐻)}+sup_{−𝑁≤𝑘<𝑘+2𝑟≤0}𝜑_𝑘+2𝑟−𝜑_𝑘𝐸(−𝑘)^𝛼(2𝑟)^−𝛼,respectively. (2.6)

Theorem 2.3. Nonlocal boundary value problem (1.3) is stable in 𝐶([−1,1]_𝜏,𝐻) space.

Proof. By [22], we have 𝑢_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝐻)}≤𝑀₃(𝛿)𝑔^𝜏_{𝐶([0,1]𝜏,𝐻)}+𝜉_𝐻+𝜓_𝐻, (2.7) for the solution of the following boundary value problem: −𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1+𝐴𝑢_𝑘=𝑔_𝑘,𝑔_𝑘=𝑔𝑡_𝑘,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁−1,₀=𝜉,𝑢_𝑁=𝜓. (2.8) By [24], we have 𝑢_𝑘⁰_{−𝑁𝐶([−1,0]𝜏,𝐻)}≤𝑀₄(𝛿)𝑓^𝜏_{𝐶([−1,0]𝜏,𝐻)}+𝜉_𝐻 (2.9) for the solution of an inverse Cauchy difference problem: 𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1−𝜏𝐼+2𝐴𝐴𝑢_𝑘−1=𝜏𝐼+2𝐴𝑓_𝑘,−(𝑁−1)≤𝑘≤0,𝑢₀=𝜉. (2.10) Then, the proof of Theorem 2.3 is based on stability inequalities (2.7) and (2.9) and on the following estimates: 𝜉_𝐻≤𝑀₅(𝛿)𝑓^𝜏_{𝐶([−1,0]𝜏,𝐻)}+𝑔^𝜏_{𝐶([0,1]𝜏,𝐻)}+𝜑_𝐻,𝜓_𝐻≤𝑀₆(𝛿)𝑓^𝜏_{𝐶([−1,0]𝜏,𝐻)}+𝑔^𝜏_{𝐶([0,1]𝜏,𝐻)}+𝜑_𝐻, (2.11) for the solution 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 𝑓₀,𝑓₋₁,𝑔₁∈𝐷(𝐼+𝜏𝐵). Then, for the solution of difference problem (1.3), the following almost coercivity inequality holds: 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−11^𝑁−1_{𝐶([0,1]𝜏,𝐻)}+𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1⁰_{−𝑁+1𝐶([−1,0]𝜏,𝐻)}+𝐴𝑢_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝐻)}+𝜏𝐼+2𝐴𝐴𝑢_𝑘−1⁰_{−𝑁+1𝐶([−1,0]𝜏,𝐻)}≤𝑀₇(𝛿1)minln𝜏,1+ln‖𝐴‖_𝐻→𝐻𝑓^𝜏_{𝐶([−1,0]𝜏,𝐻)}+𝑔^𝜏_{𝐶([0,1]𝜏,𝐻)}+𝐴𝜑_𝐻+(𝐼+𝜏𝐵)𝑓_0𝐻+(𝐼+𝜏𝐵)𝑔_1𝐻+(𝐼+𝜏𝐵)𝑓_−1𝐻. (2.12)

Proof. We have 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−11^𝑁−1_{𝐶([0,1]𝜏,𝐻)}+𝐴𝑢_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝐻)}≤𝑀₈(𝛿1)minln𝜏,1+ln‖𝐴‖_𝐻→𝐻𝑔^𝜏_{𝐶([0,1]𝜏,𝐻)}+𝐴𝜉_𝐻+𝐴𝜓_𝐻, (2.13) for the solution of boundary value problem (2.8) (see [22]), and we get 𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1⁰_{−𝑁+1𝐶([−1,0]𝜏,𝐻)}+𝜏𝐼+2𝐴𝐴𝑢_𝑘−1⁰_{−𝑁+1𝐶([−1,0]𝜏,𝐻)}≤𝑀₉(𝛿1)minln𝜏,1+ln‖𝐴‖_𝐻→𝐻𝑓^𝜏_{𝐶([−1,0]𝜏,𝐻)}+𝐴𝜉_𝐻, (2.14) 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: 𝐴𝜉_𝐻≤𝑀₁₀(𝛿)𝐴𝜑_𝐻+(𝐼+𝜏𝐵)𝑓_0𝐻1+minln𝜏,1+ln‖𝐴‖_𝐻→𝐻×𝑓^𝜏_{𝐶([−1,0]𝜏,𝐻)}+𝑔^𝜏_{𝐶([0,1]𝜏,𝐻)},𝐴𝜓_𝐻≤𝑀₁₁(𝛿)𝐴𝜑_𝐻+(𝐼+𝜏𝐵)𝑓_0𝐻1+minln𝜏,1+ln‖𝐴‖_𝐻→𝐻×𝑓^𝜏_{𝐶([−1,0]𝜏,𝐻)}+𝑔^𝜏_{𝐶([0,1]𝜏,𝐻)} (2.15) 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.

Theorem 2.5. Let assumptions of Theorem 2.5 be satisfied. Then, boundary value problem (1.3) is well-posed in Hölder spaces 𝐶^𝛼_0,1([−1,1]_𝜏,𝐻),  and  𝐶^𝛼_0,1([−1,1]_𝜏,𝐻), and the following coercivity inequalities hold: 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−11^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,H)}+𝐴𝑢_𝑘1^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝜏𝐼+2𝐴𝐴𝑢_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐻)}≤𝑀₁₂(𝛿1)𝛼()𝑓1−𝛼^{𝜏𝐶𝛼0([−1,0]𝜏,𝐻)}+𝑔^𝜏_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝐴𝜑_𝐻+(𝐼+𝜏𝐵)𝑓_0𝐻+(𝐼+𝜏𝐵)𝑔_1𝐻+(𝐼+𝜏𝐵)𝑓_−1𝐻,𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−11^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+𝐴𝑢_𝑘1^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝜏𝐼+2𝐴𝐴𝑢_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐻)}≤𝑀₁₃(𝛿1)𝛼()𝑓1−𝛼^𝜏𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+𝑔^𝜏_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝐴𝜑_𝐻+(𝐼+𝜏𝐵)𝑓_0𝐻+(𝐼+𝜏𝐵)𝑔_1𝐻+(𝐼+𝜏𝐵)𝑓_−1𝐻. (2.16)

Proof. By [22, 24], we have 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−11^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝐴𝑢_𝑘1^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐻)}≤𝑀₁₄(𝛿1)𝛼()𝑔1−𝛼^{𝜏𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝐴𝜉_𝐻+𝐴𝜓_𝐻, (2.17) for the solution of boundary value problem (2.8), and 𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+𝜏𝐼+2𝐴𝐴𝑢_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐻)}≤𝑀₁₅(𝛿1)𝛼()𝑓1−𝛼^{𝜏𝐶𝛼0([−1,0]𝜏,𝐻)}+𝐴𝜉_𝐻,𝜏⁻¹𝑢_𝑘−𝑢_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+𝜏𝐼+2𝐴𝐴𝑢_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐻)}≤𝑀₁₆(𝛿1)𝛼()𝑓1−𝛼^𝜏𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+𝐴𝜉_𝐻 (2.18)(2.19) 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: 𝐴𝜉_𝐻≤𝑀₁₇(𝛿1)𝛼()𝑓1−𝛼^𝜏𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+𝑔^𝜏_{𝐶𝛼0,1([0,1]𝜏,𝐻)}+𝐴𝜑_𝐻+(𝐼+𝜏𝐵)𝑓_0𝐻+(𝐼+𝜏𝐵)𝑔_1𝐻+(𝐼+𝜏𝐵)𝑓_−1𝐻,𝐴𝜓_𝐻≤𝑀₁₈(𝛿1)𝛼(1−𝛼)‖𝑓^𝜏‖𝐶^𝛼_{0([−1,0]𝜏,𝐻)}+‖𝑔^𝜏‖_𝐶^𝛼_{0,1([0,1]𝜏,𝐻)}+‖𝐴𝜑‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₀‖_𝐻+‖(𝐼+𝜏𝐵)𝑔₁‖_𝐻+‖(𝐼+𝜏𝐵)𝑓₋₁‖_𝐻 (2.20) 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.

3. 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: −𝑢_𝑡𝑡−^𝑛_𝑟=1𝑎_𝑟(𝑥)𝑢_{𝑥𝑟𝑥𝑟}()𝑢=𝑔𝑡,𝑥,0<𝑡<1,𝑥∈Ω,_𝑡+^𝑛_𝑟=1𝑎_𝑟(𝑥)𝑢_{𝑥𝑟𝑥𝑟}()𝑢()𝑢()==𝑓𝑡,𝑥,−1<𝑡<0,𝑥∈Ω,𝑡,𝑥=0,𝑥∈𝑆,−1≤𝑡≤1,1,𝑥^𝐽_𝑖=1𝛼_𝑖𝑢𝜆_𝑖,𝑥+𝜑(𝑥),^𝐽_𝑖=1𝛼_𝑖≤1,−1≤𝜆₁<𝜆₂<⋯<𝜆_𝑖<⋯<𝜆_𝐽𝑢()()≤0,0+,𝑥=𝑢0−,𝑥,𝑢_𝑡(0+,𝑥)=𝑢_𝑡(0−,𝑥),𝑥∈Ω, (3.1) 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: Ω_ℎ=𝑥=𝑥_𝑚=ℎ₁𝑚₁,…,ℎ_𝑛𝑚_𝑛,𝑚=𝑚₁,…,𝑚_𝑛,0≤𝑚_𝑟≤𝑁_𝑟,ℎ_𝑟𝑁_𝑟,Ω=1,𝑟=1,…,𝑛_ℎ=Ω_ℎ∩Ω,𝑆_ℎ=Ω_ℎ∩𝑆. (3.2)

We introduce the Hilbert spaces 𝐿_2ℎ=𝐿₂(Ω_ℎ),𝑊¹_2ℎ=𝑊¹₂(Ω_ℎ), and 𝑊²_2ℎ=𝑊²₂(Ω_ℎ) of the grid functions 𝜑^ℎ(𝑥)={𝜑(ℎ₁𝑚₁,…,ℎ_𝑛𝑚_𝑛)} defined on Ω_ℎ, equipped with the following norms: 𝜑^ℎ_𝐿2ℎ=_𝑥∈Ωℎ𝜑^ℎ(𝑥)²ℎ₁⋯ℎ_𝑛^1/2,𝜑^ℎ_𝑊12ℎ=𝜑^ℎ_𝐿2ℎ+_𝑥∈Ωℎ^𝑛_𝑟=1(𝜑^ℎ)_𝑥𝑟²ℎ₁⋯ℎ_𝑛^1/2,𝜑^ℎ_𝑊22ℎ=𝜑^ℎ_𝐿2ℎ+_𝑥∈Ωℎ^𝑛_𝑟=1(𝜑^ℎ)_𝑥𝑟²ℎ₁⋯ℎ_𝑛^1/2+_𝑥∈Ωℎ^𝑛_𝑟=1(𝜑^ℎ)_{𝑥𝑟𝑥𝑟,𝑚𝑟}²ℎ₁⋯ℎ_𝑛^1/2. (3.3)

To the differential operator 𝐴 generated by problem (3.1), we assign the difference operator 𝐴^𝑥_ℎ by formula 𝐴^𝑥_ℎ𝑢^ℎ=−^𝑛_𝑟=1𝑎_𝑟(𝑥)𝑢^ℎ_{𝑥𝑟𝑥𝑟,𝑚𝑟} (3.4) 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: −𝑑²𝑢^ℎ(𝑡,𝑥)𝑑𝑡²+𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑔^ℎ(𝑡,𝑥),0<𝑡<1,𝑥∈Ω_ℎ,𝑑𝑢^ℎ(𝑡,𝑥)𝑑𝑡−𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥),−1<𝑡<0,𝑥∈Ω_ℎ,𝑢^ℎ(1,𝑥)=^𝑛_𝑘=1𝛼_𝑘𝑢^ℎ𝜆_𝑘,𝑥+𝜑^ℎ(𝑥),^𝑛_𝑘=1𝛼_𝑘≤1,𝑥∈Ω_ℎ,𝑢^ℎ(0+,𝑥)=𝑢^ℎ(0−,𝑥),𝑑𝑢^ℎ(0+,𝑥)=𝑑𝑡𝑑𝑢^ℎ(0−,𝑥)𝑑𝑡,𝑥∈Ω_ℎ, (3.5) 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]): −𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)=𝑔^ℎ_𝑘(𝑥),𝑔^ℎ_𝑘(𝑥)=𝑔^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,𝑥∈Ω_ℎ,𝑢^ℎ_𝑘(𝑥)−𝑢^ℎ_𝑘−1(𝑥)𝜏−𝐴^𝑥_ℎ+𝜏2𝐴^𝑥_ℎ²𝑢^ℎ_𝑘−1(𝑥)=𝜏𝐼+2𝐴^𝑥_ℎ𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘−1/2,𝑥,𝑡_𝑘−1/2=1𝑘−2𝜏,−𝑁+1≤𝑘≤0,𝑥∈Ω_ℎ,−𝑢^ℎ₂(𝑥)+4𝑢^ℎ₁(𝑥)−3𝑢^ℎ₀(𝑥)=3𝑢^ℎ₀(𝑥)−4𝑢^ℎ₋₁(𝑥)+𝑢^ℎ₋₂(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ_𝑁(𝑥)=^𝐽_𝑘=1𝛼_𝑖𝑢^ℎ_{𝑖[𝜆/𝜏]}(𝑥)+𝜆_𝑘−𝜆_𝑖𝜏𝜏𝑓^ℎ_{𝑖[𝜆/𝜏]}+𝐴^𝑥_ℎ𝑢^ℎ_{𝑖[𝜆/𝜏]}(𝑥)+𝜑^ℎ(𝑥),𝑥∈Ω_ℎ. (3.6)

Theorem 3.1. Let 𝜏 and |ℎ|=ℎ²₁+⋯+ℎ²_𝑛 be sufficiently small positive numbers. Then, solutions of difference scheme (3.6) satisfy the following stability and almost coercivity estimates: 𝑢^ℎ_𝑘^𝑁−1_{−𝑁𝐶([−1,1]𝜏,𝐿2ℎ)}≤𝑀₁₉(𝛿)𝑓^ℎ_𝑘⁻¹_{−𝑁+1𝐶([−1,0]𝜏,𝐿2ℎ)}+𝑔^ℎ_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝐿2ℎ)}+𝜑^ℎ_𝐿2ℎ,𝜏⁻²𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−11^𝑁−1_{𝐶([0,1]𝜏,𝐿2ℎ)}+𝑢^ℎ_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝑊22ℎ)}+𝜏⁻¹𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1⁰_{−𝑁+1𝐶([−1,0]𝜏,𝐿2ℎ)}+𝑢^ℎ_𝑘−1⁰_{−𝑁+1𝐶([−1,0]𝜏,𝑊22ℎ)}≤𝑀₂₀(𝛿)𝑓^ℎ_0𝐿2ℎ+𝑓^ℎ_−1𝐿2ℎ+𝑔^ℎ_1𝐿2ℎ+𝜑^ℎ_𝑊22ℎ+𝜏𝑓^ℎ_0𝑊12ℎ+𝜏𝑓^ℎ_{−1𝑊12ℎ}+𝜏𝑔^ℎ_1𝑊12ℎ1+ln|ℎ|𝑓𝜏+^{ℎ𝑘−1−𝑁+1𝐶([−1,0]}_𝜏^,𝐿_2ℎ⁾+𝑔^ℎ_𝑘1^𝑁−1_{𝐶([0,1]𝜏,𝐿2ℎ)}. (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 1minln𝜏,1+ln𝐴^𝑥_{ℎ𝐿2ℎ→𝐿2ℎ}≤𝑀₂₁(𝛿)1ln|ℎ|,𝜏+ (3.8) 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_ℎ𝑢^ℎ(𝑥)=𝜔^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ(𝑥)=0,𝑥∈𝑆_ℎ, (3.9) the following coercivity inequality holds [25]: ^𝑛_𝑟=1𝑢^ℎ_{𝑥𝑟𝑥𝑟,𝑚𝑟𝐿2ℎ}≤𝑀₂₂(𝛿)𝜔^ℎ_𝐿2ℎ. (3.10)

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: 𝜏⁻²𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−11^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐿2ℎ)}+𝜏⁻¹𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐿2ℎ)}+𝑢^ℎ_𝑘1^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝑊22ℎ)}+𝑢^ℎ_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝑊}²_2ℎ)≤𝑀₂₃(𝛿)𝜑^ℎ_𝑊22ℎ+𝜏𝑓^ℎ_0𝑊12ℎ+𝜏𝑓^ℎ_{−1𝑊12ℎ}+𝜏𝑔^ℎ_1𝑊12ℎ+1𝛼()𝑓1−𝛼^{ℎ𝑘−1−𝑁+1𝐶𝛼}₀^([−1,0]_𝜏^,𝐿_2ℎ⁾+𝑔^ℎ_𝑘1^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐿2ℎ)},𝜏⁻²𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_𝑘−11^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐿2ℎ)}+𝑢^ℎ_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝑊}²_2ℎ)+𝜏⁻¹𝑢^ℎ_𝑘−𝑢^ℎ_𝑘−1⁰_−𝑁+1𝐶^𝛼_{0([−1,0]𝜏,𝐿2ℎ)}+𝑢^ℎ_𝑘1^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝑊22ℎ)}≤𝑀₂₄(𝛿)𝜑^ℎ_𝑊22ℎ+𝜏𝑓^ℎ_0𝑊12ℎ+𝜏𝑓^ℎ_{−1𝑊12ℎ}+𝜏𝑔^ℎ_1𝑊12ℎ+1𝛼()𝑓1−𝛼^{ℎ𝑘−1−𝑁+1}𝐶^𝛼_{0([−1,0]𝜏,𝐿2ℎ)}+𝑔^ℎ_𝑘1^𝑁−1_{𝐶𝛼0,1([0,1]𝜏,𝐿2ℎ)}. (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).