The Theory and Computation of the Semi-Linear Reaction–Diffusion Equation with Dirichlet Boundaries

In this article, we study the semi-linear two-dimensional reaction–diffusion equation with Dirichlet boundaries. A reliable numerical scheme is designed, coupling the nonstandard finite difference method in the time together with the Galerkin in combination with the compactness method in the space variables. The aforementioned equation is analyzed to show that the weak or variational solution exists uniquely in specified space. The a priori estimate obtained from the existence of the weak or variational solution is used to show that the designed scheme is stable and converges optimally in specified norms. Furthermore, we show that the scheme preserves the qualitative properties of the exact solution. Numerical experiments are presented with a carefully chosen example to validate our proposed theory.