Abstract: The adaptive finite element methods are very effective for solving partial differential equations in scientific researches and engineering designs.By using these methods the best possible results can be obtained at less computational costs.A posteriori error estimates serve as a key to realize the adaptive finite element computation.This paper surveys the progress in the adaptive finite element methods and a posteriori error estimates for solving elliptic equations,parabolic equations and hyperbolic equations.

Key words: Adaptive finite element methods, a posteriori error estimates, elliptic equations, parabolic equations, hyperbolic equations