Solving Biharmonic Eigenvalue Problem With Navier Boundary Condition Via Poisson Solvers On Non-Convex Domains

TL;DR

This paper introduces a Poisson solver-based method for solving the biharmonic eigenvalue problem with Navier boundary conditions on non-convex domains, avoiding spurious eigenvalues and providing error estimates and convergence results.

Contribution

It develops a novel algorithm decomposing the biharmonic problem into three Poisson equations, ensuring accurate eigenfunctions and eigenvalues on non-convex domains.

Findings

Avoids spurious eigenvalues in non-convex domains

Achieves a convergence rate of O(h^{2α}) for eigenvalues and eigenfunctions

Numerical results show an O(h^2) convergence rate even on non-convex polygons

Abstract

It is well known that the usual mixed method for solving the biharmonic eigenvalue problem by decomposing the operator into two Laplacians may generate spurious eigenvalues on non-convex domains. To overcome this difficulty, we adopt a recently developed mixed method, which decomposes the biharmonic equation into three Poisson equations and still recovers the original solution. Using this idea, we design an efficient biharmonic eigenvalue algorithm, which contains only Poisson solvers. With this approach, eigenfunctions can be confined in the correct space and thereby spurious modes in non-convex domains are avoided. A priori error estimates for both eigenvalues and eigenfunctions on quasi-uniform meshes are obtained; in particular, a convergence rate of $\mathcal{O}({h}^{2\alpha})$ ($ 0<\alpha<\pi/\omega$, $\omega > \pi$ is the angle of the reentrant corner) is proved for the linear finite element. Surprisingly, numerical evidence demonstrates a $\mathcal{O}({h}^{2})$ convergent rate for the quasi-uniform mesh with the regular refinement strategy even on non-convex polygonal domains.