A new nonlinear Galerkin finite element method for the computation of reaction diffusion equations

To get the stationary patterns in the approximation of reaction diffusion systems, the computation of evolution equations on large intervals of time is required. In this paper, a new nonlinear Galerkin method based on finite element discretization is presented for the approximation of reaction diffusion equations on large intervals. The new scheme is based on two different finite element spaces defined respectively on one subspace of lower-degree shape function and one of higher-degree shape function. Nonlinearity and time dependence are both treated on the lower-degree space and only a fixed stationary equation needs to be solved on the higher-degree space at each time level. The proof of the stability and convergence of the method is presented in one dimension. Numerical solutions of some well-studied reaction-diffusion systems are presented to demonstrate the effectiveness of the nonlinear Galerkin method.