Iterative finite element approximation of non-monotone semilinear diffusion-reaction equations
We study iterative linearized finite element methods for the numerical approximation of semilinear elliptic boundary value problems with nonlinear reaction terms of asymptotically linear growth. Our approach reaches considerably beyond the classical theory by allowing for nonlinearities that are not necessarily monotone. We investigate a stabilized implicit-explicit (IMEX) iteration combined with first-order finite element discretizations on graded meshes that are able to resolve corner singularities in polygonal domains. To account for the reduced regularity of the analytical solutions, we establish regularity estimates in corner-weighted Sobolev spaces. The principal novelty of the paper is a non-standard analysis of the approximation properties of the discrete Galerkin limits resulting from the iterative process, which yields optimal a priori convergence estimates. Numerical experiments underline the theoretical findings.