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Fully discrete finite element approximations of the forced Fisher equation. (English) Zbl 1095.65094
From the authors’ abstract: We consider fully discrete finite element approximations of the forced Fisher equation that models 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.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
92D25 Population dynamics (general)
35Q80 Applications of PDE in areas other than physics (MSC2000)
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