Department of Mathematics, Pennsylvania State University, USA.

Academia Sinica, P.R. China.

In this talk, some local and parallel adaptive finite element algorithms for nonlinear elliptic equations in both two and three dimensions will be reported. These algorithms show that, for a solution to some nonlinear elliptic problem, low frequency components can be approximated well by a standard finite element discretization on a relatively coarse grid and high frequency components can be obtained by some linearized discretization on some local fine grid in some parallel procedure. The theoretical tools for analyzing these algorithms are some local a priori and a posteriori error estimates for finite element soultions on general shape-regular grids. Multigrid and domain decomposition techniques both play important roles in this approach.