A Type Of Multigrid Method Based On The Fixed-shift Inverse Iteration For The Steklov Eigenvalue Problem

Steklov eigenvalue problems in which the eigenvalue parameter appears in the boundary condition have important physical background.Thus numerical methods for Steklov eigenvalue problems have gradually become hot topics focused on by scholars.In the numerical approximation of the partial differential equations,the adaptive procedures based on a posteriori error estimates,due to the less computational cost and time,are the mainstream direction and have gained an enormous importance.Combining the finite element method and the fixed-shift inverse iteration,a type of multigrid discretizations based on the fixed-shift inverse iteration for the Steklov eigenvalue problem is proposed in this paper.With this method,the solution of the Steklov eigenvalue problem is reduced to the solution of an eigenvalue problem on a much coarser grid V_H and the solution of a series of linear algebraic equations on finer and finer grids V_hi.This paper studies in depth the a priori error estimates and the residual type a posteriori error estimates,and prove the global reliability and local efficiency of the posteriori error estimator.In addition,based on the a posteriori error estimates,we design a new adaptive algorithm of fixed-shift inverse iteration type.This algorithm not only has less computational complexity,but also avoid the computational difficulty caused by the nearly singular algebraic system.And thus has more efficiency.Finally,comparing the performance of three types of different adaptive algorithms,numerical experiments with MATLAB program on the unit square,L-shape domain and rhombic slit domain are presented to validate the efficiency of our method.