Cascadic adaptive finite element method for nonlinear eigenvalue problem based on complementary approach

Abstract

In this paper, a type of cascadic adaptive finite element method for nonlinear eigenvalue problem is proposed based on the complementary approach. In this new scheme, we only need to do some smoothing steps for the linearized boundary value problems on a series of adaptive finite element spaces and solve some nonlinear eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the nonlinear eigenvalue problems on each adaptive space directly. Besides, the complementary error estimate for nonlinear eigenvalue problem will be introduced in our paper. This estimate can not only provide an accurate error estimate for the nonlinear eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm. Some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper.