A High-order Arbitrary Lagrangian-Eulerian Virtual Element Method for Convection-Diffusion Problems

TL;DR

This paper introduces a high-order virtual element method within an Arbitrary Lagrangian-Eulerian framework for 2D convection-diffusion problems, achieving optimal convergence on curved polygonal meshes and improving moving boundary simulations.

Contribution

It develops a novel isoparametric virtual element discretisation for ALE methods, enabling higher-order accuracy on complex geometries and integration with moving mesh algorithms.

Findings

Achieves optimal $H^1$ and $L^2$ convergence rates.

Successfully applies to moving boundary problems with higher-order accuracy.

Validates the method through numerical experiments.

Abstract

A virtual element discretisation of an Arbitrary Lagrangian-Eulerian method for two-dimensional convection-diffusion equations is proposed employing an isoparametric Virtual Element Method to achieve higher-order convergence rates on curved edged polygonal meshes. The proposed method is validated with numerical experiments in which optimal $H^1$ and $L^2$ convergence are observed. This method is then successfully applied to an existing moving mesh algorithm for implicit moving boundary problems in which higher-order convergence is achieved.