Analysis of two-grid method for semi-linear elliptic equations by new mixed finite element scheme

This article considers two-grid stable mixed finite element method based on the less regularity of flux (velocity) in practice for the semi-linear elliptic equations approximated by the P0-P1P0-P1 (velocity–pressure) pair which satisfies the inf-sup condition. This method involves solving a small semi-linear system on a coarse mesh with mesh size H   and a linear system problem on a fine mesh with mesh size H=O(h), which can still maintain the asymptotically optimal accuracy. It provides an approximate solution with the convergence rate of the same order as the usual mixed finite element solution solving the semi-linear elliptic equations on a fine mesh. Hence, The two-grid scheme can reduce the computational cost. Finally, numerical tests confirm the theoretical results of the present method.