THE FINITE-ELEMENT APPROXIMATION OF A COUPLED REACTION-DIFFUSION PROBLEM WITH NON-LIPSCHITZ NONLINEARITIES

A coupled semilinear elliptic problem modelling an irreversible, isoth ermal chemical reaction is introduced, and discretised using the usual piecewise linear Galerkin finite element approximation. An interestin g feature of the problem is that a reaction order of less than one giv es rise to a ''dead core'' region. Initially, one reactant is assumed to be acting as a catalyst and is kept constant. It is shown that erro r bounds previously obtained for a scheme involving numerical integrat ion can be improved upon by considering a quadratic regularisation of the nonlinear term. This technique is then applied to the full coupled problem, and optimal H-1 and L(infinity) error bounds are proved in t he absence of quadrature. For a scheme involving numerical integration , bounds similar to those obtained for the catalyst problem are shown to hold.