On Galerkin Methods in Semilinear Parabolic Problems

Abstract

We consider the approximate solution by Galerkin’s method of the initial boundary value problem for the semilinear equation $u_t = \Delta u + f(u)$ in $\Omega \times [0,T]$ with $u = 0$ on $\partial \Omega \times [0,T]$. Optimal order error estimates in $L_2 $ and $H^1 $ are derived under various assumptions on f and the subspaces in which the approximation is sought, and for various choices of the Galerkin procedure.