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–diffusion equations using an expanded mixed finite 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 fine 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 fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^1/3). As a result, solving such a large class of non‐linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd.