An accurate a posteriori error estimator for semilinear Neumann problem and its applications

摘要

In this paper, a type of accurate a posteriori error estimator is proposed for semilinear Neumann problem, which provides an asymptotic exact estimate for the finite element approximate solution. As its applications, we design two types of cascadic adaptive finite element methods for semilinear Neumann problem based on the proposed a posteriori error estimator. The first scheme is based on the Newton iteration, which needs to solve a linearized boundary value problem by some smoothing steps on each adaptive space. The second scheme is based on the multilevel correction method, which contains some smoothing steps for a linearized boundary value problem on each adaptive space and a solving step for semilinear Neumann equation on a low dimensional space. In addition, the proposed a posteriori error estimator provides the strategy to refine mesh and control the number of smoothing steps for both of the cascadic adaptive methods. Some numerical examples are presented to validate the efficiency of the proposed algorithms in this paper.