A virtual element method on polyhedral meshes for the sixth-order elliptic problem

TL;DR

This paper develops and analyzes a virtual element method on polyhedral meshes for solving sixth-order elliptic problems, providing convergence analysis and numerical validation.

Contribution

It introduces a novel virtual element discretization for sixth-order elliptic problems on polyhedral meshes with convergence and error estimates.

Findings

Method achieves optimal convergence rates.

Numerical tests confirm theoretical error estimates.

Effective for complex polyhedral geometries.

Abstract

In this work we analyze a virtual element method on polyhedral meshes for solving the sixth-order elliptic problem with simply supported boundary conditions. We apply the Ciarlet-Raviart arguments to introduce an auxiliary unknown $\sigma:=-\Delta^2 u$ and to search the main uknown $u$ in the $H^2\cap H_0^1$ Sobolev space. The virtual element discretization is well possed on a $C^1\times C^0$ virtual element spaces. We also provide the convergence and error estimates results. Finally, we report a series of numerical tests to verify the performance of numerical scheme.