Nonlinear Galerkin method for reaction–diffusion systems admitting invariant regions

The article presents an analysis of the nonlinear Galerkin method applied to a system of reaction–diffusion equations. If the system admits a bounded invariant region, it is possible to demonstrate the convergence of the approximate solutions to the weak solution of the system. The proof is based on the compactness technique. It is performed for arbitrary ratio of dimensions of the approximation space and of the correction space used in the nonlinear Galerkin method. This fact, generalizing the previously published results, is important for the practical use of the method and allows optimization of the CPU-time consumption of the algorithm. The method is applied to the well-known Brusselator system for which we present an overview of the computational results and our experience with the numerical method used.