A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations

We present a scheme for solving two-dimensional semilinear reaction–di usion equations using an expanded mixed nite element method. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method. The solution of a non-linear system on the ne space is reduced to the solution of two small (one linear and one non-linear) systems on the coarse space and a linear system on the ne space. It is shown that the coarse grid can be much coarser than the ne grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H =O(h1=3). As a result, solving such a large class of non-linear equation will not be much more di cult than solving one single linearized equation