An efficient adaptive multigrid method for the elasticity eigenvalue problem

Abstract

This paper aims to introduce a novel adaptive multigrid method for the elasticity eigenvalue problem. Different from the developing adaptive algorithms for the elasticity eigenvalue problem, the proposed approach transforms the elasticity eigenvalue problem into a series of boundary value problems in the adaptive spaces and some small-scale elasticity eigenvalue problems in a low-dimensional space. As our algorithm avoids solving large-scale elasticity eigenvalue problems, which is time-consuming, and the boundary value problem can be solved efficiently by the adaptive multigrid method, our algorithm can evidently improve the overall solving efficiency for the elasticity eigenvalue problem. Meanwhile, we present a rigorous theoretical analysis of the convergence and optimal complexity. Finally, some numerical experiments are presented to validate the theoretical conclusions and verify the numerical efficiency of our approach.