Nonlinear Galerkin finite element method applied to the system of reaction-diffusion equations in one space dimension

We study the finite-element nonlinear Galerkin method in one spatial dimension and its application to the numerical solution of nontrivial dynamics in selected reaction-diffusion systems. This method was suggested as well adapted for the long-term integration of evolution equations and is studied as an alternative to the commonly used numerical approaches. The proof of the convergence of the method applied to a particular class of reaction-diffusion systems is presented. Computational properties are illustrated by results of numerical simulations. We performed the measurement of the experimental order of convergence and the computational efficiency in comparison to the usual finite-difference method.