Two-grid mixed finite element method for nonlinear hyperbolic equations

In this paper, a combination method of mixed finite element method and two-grid scheme is constructed for solving numerically the two-dimensional nonlinear hyperbolic equations. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h≪H. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes H=O(h12) in the first algorithm and H=O(h14) in the second algorithm, respectively. Error estimates are derived in detail. Finally, two numerical examples are carried out to verify the accuracy and efficiency of the presented method.