A Two-Grid Finite Element Method for a Second-Order Nonlinear Hyperbolic Equation

Abstract

We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of size $H$. The nonlinearities are expanded about the coarse grid solution on the fine gird of size $h$. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with the $H^1$-norm $O(h+H^2)$ for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfy $h=O(H^2)$.