Fully discrete finite element approximations of the forced Fisher equation

We consider fully discrete finite element approximations of the forced Fisher equation that models the dynamics of gene selection/migration for a diploid population with two available alleles in a multidimensional habitat and in the presence of an artificially introduced genotype. Finite element methods are used to effect spatial discretization and a nonstandard backward Euler method is used for the time discretization. Error estimates for the fully discrete approximations are derived by applying the Brezzi–Rappaz–Raviart theory for the approximation of a class of nonlinear problems. The approximation schemes and error estimates are applicable under weaker regularity hypotheses than those that are typically assumed in the literature. The algorithms and analyses, although presented in the concrete setting of the forced Fisher equation, also apply to a wide class of semilinear parabolic partial differential equations.