A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations

Abstract

A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid size $H$. The nonlinearities are expanded about the coarse grid solution on a fine gird of size $h$, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with the $H^1$-norm $O(h+H^2)$ which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfy $h=O(H^2)$. An example is also given to illustrate the theoretical results.