Full discretization of the porous medium/fast diffusion equation based on its very weak formulation

Abstract

SKA ‡ Abstract. The very weak formulation of the porous medium/fast diffusion equation yields an evolution problem in a Gelfand triple with the pivot space H 1. This allows to employ methods of the theory of monotone operators in order to study fully discrete approximations combining a Galerkin method (including conforming finite element methods) with the backward Euler scheme. Convergence is shown even for rough initial data and right-hand sides. The theoretical results are illustrated for the piecewise constant finite element approximation of the porous medium equation with the�-distribution as initial value. As a byproduct, L p -stability of theH 1 -orthogonal projection onto the space of piecewise constant functions is shown. ut − �(juj p−2 u) = f; p >1; u= u(x;t); u(�;0) = u0; (x;t) 2 � (0;T); whereR d is a nice bounded domain, −� is the Laplace operator acting on the spatial variables, u0 are given initial data, f is a given right-hand side and we assume some boundary condition on the boundary of . We immediately see that for p = 2 this is the classical heat equation. For p > 2, it is referred to as the porous medium equation. In this case u is the density of the gas at a given point and time. For 1 2, in the one dimensional case, the solution is given by (5.2). We will later use this in numerical experiments. The solution is also known in higher dimensions, see again Vazquez (22, Chapter 4). Throughout the article, we focus on the following generalization of the above