A two-grid algorithm based on expanded mixed element discretizations for strongly nonlinear elliptic equations

An expanded mixed element method is presented to solve a strongly nonlinear elliptic problem. Existence and uniqueness of approximation solution are analyzed. Error estimates in Lq and H−s norms are also obtained in this paper. To solve the resulting nonlinear system of equations efficiently, we use a two-grid algorithm to decompose the nonlinear system into a small nonlinear system on a coarse grid with mesh size H and a linear system on a fine grid with mesh size h. It’s shown that the approximation still achieves asymptotically optimal as long as the mesh sizes satisfy H=O(h1/2). Some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed numerical algorithm.