A Multi-level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

TL;DR

This paper introduces a multi-level mixed element method for solving the biharmonic eigenvalue problem, achieving optimal convergence and computational efficiency through nested discretizations and finite element schemes.

Contribution

It presents a novel multi-level mixed finite element scheme with optimal convergence and computational cost for biharmonic eigenvalue problems.

Findings

Optimal convergence rate demonstrated

Efficient multi-level scheme verified numerically

Nested discretizations improve computational performance

Abstract

In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization. Then, we construct multi-level finite element schemes by implementing the algorithm as in [33] to the nested discretizations on series of nested grids. The multi-level mixed scheme for biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.