A two-grid stabilized mixed finite element method for semilinear elliptic equations

In this paper, we present a two-grid stabilized mixed finite element method for the two-dimensional semilinear elliptic equations. The method combines the two grid scheme with a stabilized mixed finite element method which is based on pressure projection stabilization method by using the lowest equal-order pair for the velocity and pressure. The two-grid method consists of solving a small semilinear system on the coarse mesh and then solving a linear system on the fine mesh. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h12). Error estimates are derived for the algorithm of the two-grid method. Finally, we give some numerical experiments to verify the efficiency and the theoretical results of the proposed method.