$C^1$ virtual element methods on polygonal meshes with curved edges

TL;DR

This paper introduces a new $C^1$-conforming virtual element method for solving biharmonic problems on curved domains, providing optimal error estimates and validating results with numerical experiments.

Contribution

It develops a novel $C^1$ virtual element method of arbitrary order for curved domains, with rigorous analysis and validation.

Findings

Optimal error estimates in the energy norm

Method effectively handles curved boundaries and interfaces

Numerical experiments confirm theoretical convergence

Abstract

In this work we design a novel $C^1$-conforming virtual element method of arbitrary order $k \geq 2$, to solve the biharmonic problem on a domain with curved boundary and internal curved interfaces in two dimensions. By introducing a suitable stabilizing form, we develop a rigorous interpolation, stability and convergence analysis obtaining optimal error estimates in the energy norm. Finally, we validate the theoretical findings through numerical experiments.