Regular and exponential convergence of difference schemes for the heat-conduction equation

The mathematical model for the heat-conduction equation has several special characteristic properties. In this paper, we examine the following property. By increasing time, the solution of the problem tends to the solution of the corresponding elliptic problem. Moreover, the convergence takes place without oscillation and the convergence rate in l2-norm is the same as the convergence rate of the exponential function to zero. Applying some numerical process, it is not less important to require the preservation of the discrete analogues of the basic qualitative properties of the continuous solution at certain fixed numerical solution (or at all of them). We introduce the (σ, θ)-method which is the generalization both of the well-known Galerkin linear finite element method and the finite difference method and formulate the conditions of the preservation of the regular and exponential convergence. © 1999 Elsevier Science Ltd. All rights reserved.