$C^1$ Virtual Element Method on polyhedral meshes

TL;DR

This paper introduces a $C^1$ Virtual Element Method for 3D linear elliptic fourth order problems on polyhedral meshes, addressing limitations of standard finite elements with a novel low-order scheme validated by numerical tests.

Contribution

It develops the first $C^1$ Virtual Element Method in three dimensions for fourth order problems, with a simple degrees of freedom scheme and error estimates.

Findings

The method achieves optimal interpolation error estimates.

Numerical tests confirm the practical effectiveness of the scheme.

The approach extends to higher orders with brief generalization.

Abstract

The purpose of the present paper is to develop $C^1$ Virtual Elements in three dimensions for linear elliptic fourth order problems, motivated by the difficulties that standard conforming Finite Elements encounter in this framework. We focus the presentation on the lowest order case, the generalization to higher orders being briefly provided in the Appendix. The degrees of freedom of the proposed scheme are only 4 per mesh vertex, representing function values and gradient values. Interpolation error estimates for the proposed space are provided, together with a set of numerical tests to validate the method at the practical level.