Stability of a three-layer operator-difference scheme for coupled thermoelasticity problems

Abstract

Stability of a three-layer operator-difference scheme with weights, which generalizes a class of difference and projection-difference schemes for linear coupled thermoelasticity problems, is analyzed. Energy estimates for the solution and its first-order grid derivative are obtained.In this paper, a three-layer operator-difference scheme with weights is considered. The system of equations in this scheme consists of discrete analogs of coupled abstract hyperbolic and parabolic equations (see (2.1)-(2.3) below). The computational schemes considered here can be applied to a rather wide class of linear coupled thermoelasticity problems. These are: three-dimensional thermoelastic problems of both classical and moment (nonsymmetric) theories of elasticity (see, for instance, [1, § § 12.2, 13.5]), as well as problems of the theory of plates and shells produced by various kinematic (Kirchhoff-Love, Timoshenko) models, with two-and three-dimensional heat conduction equations. The problems can be discretized in space by difference or projection Galerkin methods (in particular, by finite element methods). For the scheme considered in this paper, stability estimates similar to energy estimates of the solution to an abstract Cauchy problem, such as a thermoelasticity problem presented in [2], are established for the solution and the first derivative of this solution. The major results of this paper are Theorems 4.1-4.6. These results can be used in the investigation of accuracy (by obtaining error estimates) of difference and projection-difference schemes for thermoelasticity problems. One particular result of those obtained here was used in [3] to investigate the accuracy of a projection-difference method for thermoelasticity problems.A general theory of stability of linear operator-difference schemes was developed in [4,5] and other papers (see References in [5]). Additional papers worth mentioning are [6][7][8][9][10][11]. However, a peculiarity of the scheme considered here is that obtaining of stability estimates for it by directly applying the well-known general methods seems impossible. Also, any attempts to transform the initial scheme (for instance, to a two-layer scheme) for subsequent use of general stability theorems do not hold much promise: It is more convenient to deal with the initial scheme. It should be noted that some stability conditions of three-and two-layer operator-difference schemes, as well as a method of energy inequalities presented in [4,5], are extensively used in this paper. ABSTRACT CAUCHY PROBLEMFirst, consider a Cauchy problem for the following system of abstract differential equations:(1.1) J 2 θ (t) + L 2 θ(t) + Nu (t) = f 2 (t), (1.2) u(0) = u 0 , u (0) = u 1 , θ(0) = θ 0 .