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

H. Wells

TL;DR

This work develops a high-order conservative ALE discretisation for convection-diffusion on moving domains using isoparametric Virtual Element Methods (IsoVEM) on curved polygonal meshes. By formulating moving VEM spaces and leveraging an isoparametric transformation, the method achieves optimal $O(h^k)$ convergence in the $H^1$-norm and $O(h^{k+1})$ in the $L^2$-norm, validated through comprehensive numerical experiments. The framework is extended to a velocity-based moving mesh approach, demonstrated on the Porous Medium Equation, where quadratic elements yield up to $O(h^3)$ accuracy for the solution, though higher-order discretisations encounter suboptimal rates in implicit moving-boundary settings. The results establish the practicality of high-order VEM on polygonal, curved meshes for moving-boundary problems and point to future work in 3D extensions, stability analyses, and Navier–Stokes extensions.

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.

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

TL;DR

This work develops a high-order conservative ALE discretisation for convection-diffusion on moving domains using isoparametric Virtual Element Methods (IsoVEM) on curved polygonal meshes. By formulating moving VEM spaces and leveraging an isoparametric transformation, the method achieves optimal convergence in the -norm and in the -norm, validated through comprehensive numerical experiments. The framework is extended to a velocity-based moving mesh approach, demonstrated on the Porous Medium Equation, where quadratic elements yield up to accuracy for the solution, though higher-order discretisations encounter suboptimal rates in implicit moving-boundary settings. The results establish the practicality of high-order VEM on polygonal, curved meshes for moving-boundary problems and point to future work in 3D extensions, stability analyses, and Navier–Stokes extensions.

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 and 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.
Paper Structure (31 sections, 62 equations, 2 figures, 8 tables)

This paper contains 31 sections, 62 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. The Convection-diffusion Problem
  3. The ALE Mapping
  4. The Convection-diffusion Equation
  5. Weak Formulation
  6. The Virtual Element Method
  7. A Computational Reference Mesh
  8. Polynomial Projection Operators
  9. Local & Global VEM spaces
  10. Degrees of Freedom
  11. Representation of VEM Functions
  12. Approximation of Linear and Bilinear Forms
  13. A Conservative ALE-VEM Formulation
  14. Approximating the ALE Mapping
  15. Moving Virtual Element Spaces
  16. ...and 16 more sections

Figures (2)

  • Figure 1: Solution snapshots of the convection-diffusion equation using a quadratic VEM and a reference mesh of 800 elements. The snapshots are taken at times $t=0$ (top left), $t=0.025$ (top right), $t=0.05$ (bottom left) and $t=0.075$ (bottom right).
  • Figure 2: Solution snapshots of the convection-diffusion equation using a quadratic VEM and a reference mesh of 800 elements. The snapshots are taken at times $t=0$ (top left), $t=0.025$ (top right), $t=0.05$ (bottom left) and $t=0.075$ (bottom right).

Theorems & Definitions (4)

  • Definition 3.1: The $\Pi^{\nabla}$ Operator
  • Definition 3.2: The $\Pi^{0}$ Operator
  • Definition 3.3: The $\Pi^{1}$ Operator
  • Definition 3.4: Degrees of Freedom for the Virtual Element Method