  In this paper we consider the numerical approximation of a general second
order semi-linear parabolic partial differential equation. Equations of this
type arise in many contexts, such as transport in porous media. Using finite
element method for space discretization and the exponential Rosenbrock-Euler
method for time discretization, we provide a rigorous convergence proof in
space and time under only the standard Lipschitz condition of the nonlinear
part for both smooth and nonsmooth initial solution. This is in contrast to
very restrictive assumptions made in the literature, where the authors have
considered only approximation in time so far in their convergence proofs. The
optimal orders of convergence in space and in time are achieved for smooth and
nonsmooth initial solution.
