A diffusion--modified quadrature finite element method for nonlinear reaction--diffusion equations

Abstract

In this work we propose, analyse and implement a fully discrete diffusion-modified H 1 -Galerkin method with quadrature for solving nonlinear variable diffusion coefficient reaction-diffusion equations on a rectangular region. In our least square quadrature finite element method, the trial space consists of twice continuously differentiable cubic or higher degree splines and the test space is obtained by applying the second order diffusion operator to the trial space. At each discrete time step, our algorithm requires solution of only a constant diffusion coefficient fully discrete linear problem. We prove that for sufficiently small time step-size, the scheme is stable and converges with optimal order accuracy in time and space (with H 1 or H 2 norm). Finally, we present numerical results demonstrating the accuracy of our scheme in L 2 , H 1 and H 2 norms.