Maximum norm contractivity in the numerical solution of the one-dimensional heat equation

In this paper we consider the one-dimensional heat conduction equation (Fri edmann, 1964). To the numerical solution of the problem we apply the so-cal led (sigma, theta)-method (Farago, 1996; Thomee, 1990) which unites a few n umerical methods. With the choice sigma = 0 we get the finite difference th eta-method and the choice sigma = 1/6 results in the finite element method with linear elements. The most important question is the choice of the suit able mesh-parameters. The basic condition arises from the condition of the convergence (Farago, 1996; Samarskii, 1977; Thomee, 1990). Further conditio ns can be obtained aiming at preserving some qualitative properties of the continuous problem. Some of them are the following: non-negativity, convexi ty, concavity, shape preservation and contractivity in some norms (Dekker a nd Verwer, 1984). Now we shall study the maximum norm contractivity. There are some results in the literature for the parameter choices which guarante e this property (Kraaijevanger, 1992; Samarskii, 1977; Thomee, 1990). Howev er these papers specialize only on the finite difference methods and give s ufficient conditions. We determine the necessary and sufficient conditions related to the (sigma, theta)-method. We close the paper with numerical exa mples. (C) 1999 Elsevier Science B.V. and IMACS. All rights reserved.