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The Mixed Virtual Element Discretization for highly-anisotropic problems: the role of the boundary degrees of freedom

Stefano Berrone, Stefano Scialò, Gioana Teora

TL;DR

This work analyzes the robustness of the mixed Virtual Element Method for highly anisotropic diffusion on general polytopal meshes. It introduces boundary degrees of freedom defined by $L^2([0,1])$-orthonormal moments and evaluates their impact alongside stabilization strategies, comparing monomial versus orthonormal polynomial bases for velocity and pressure spaces. Key findings show that orthonormal bases significantly improve high-$k$ conditioning and accuracy, while boundary DOFs mainly shift error curves without guaranteeing conditioning gains; the D-recipe stabilization with unit constant provides robust performance across anisotropic scenarios. The results guide practical choices for stabilizations and DOFs in anisotropic VEM applications, including challenging magnetic island-like diffusion problems.

Abstract

In this paper, we discuss the accuracy and the robustness of the mixed Virtual Element Methods when dealing with highly-anisotropic diffusion problems. In particular, we analyze the performances of different approaches which are characterized by different sets of both boundary and internal degrees of freedom in presence of a strong anisotropy of the diffusion tensor with constant or variable coefficients. A new definition of the boundary degrees of freedom is also proposed and tested.

The Mixed Virtual Element Discretization for highly-anisotropic problems: the role of the boundary degrees of freedom

TL;DR

This work analyzes the robustness of the mixed Virtual Element Method for highly anisotropic diffusion on general polytopal meshes. It introduces boundary degrees of freedom defined by -orthonormal moments and evaluates their impact alongside stabilization strategies, comparing monomial versus orthonormal polynomial bases for velocity and pressure spaces. Key findings show that orthonormal bases significantly improve high- conditioning and accuracy, while boundary DOFs mainly shift error curves without guaranteeing conditioning gains; the D-recipe stabilization with unit constant provides robust performance across anisotropic scenarios. The results guide practical choices for stabilizations and DOFs in anisotropic VEM applications, including challenging magnetic island-like diffusion problems.

Abstract

In this paper, we discuss the accuracy and the robustness of the mixed Virtual Element Methods when dealing with highly-anisotropic diffusion problems. In particular, we analyze the performances of different approaches which are characterized by different sets of both boundary and internal degrees of freedom in presence of a strong anisotropy of the diffusion tensor with constant or variable coefficients. A new definition of the boundary degrees of freedom is also proposed and tested.
Paper Structure (13 sections, 49 equations, 13 figures)

This paper contains 13 sections, 49 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The model problem
  3. The mixed Virtual Element Space
  4. The Degrees of Freedom for the velocity variable
  5. The Mixed Virtual Element Formulation of the model problem
  6. The stabilization term
  7. Numerical experiments
  8. Test 1: Boundary degrees of freedom
  9. Test 2: Anisotropic diffusion problems
  10. Effect of the anisotropy on the condition number of the global system matrix
  11. The mesh alignment and the locking phenomenon
  12. Test 3: Two Magnetic Islands
  13. Conclusions

Figures (13)

  • Figure 1: Test 1. The three concave refinements $\mathcal{T}_{h\ifstrempty{i}{}{_{i}}}^C$, $i=1,2,3$.
  • Figure 2: Test 1. Condition number of $\mathbf{K}$ vs. $k$. Left: Mesh $\mathcal{T}_{h\ifstrempty{1}{}{_{1}}}^C$. Center: Mesh $\mathcal{T}_{h\ifstrempty{2}{}{_{2}}}^C$. Right: Mesh $\mathcal{T}_{h\ifstrempty{3}{}{_{3}}}^C$.
  • Figure 5: Test 1. Behaviour of $\mathrm{err}_p$\ref{['eq:errorp']} vs. $h$. Left: $k=1$. Center: $k=3$. Right: $k=5$.
  • Figure 7: Test 2. \ref{['fig:Q5x5']}: Mesh $\mathcal{T}_{h\ifstrempty{1}{}{_{1}}}^{Q}$. \ref{['fig:QD5x5']}: Mesh $\mathcal{T}_{h\ifstrempty{1}{}{_{1}}}^{DQ}$. \ref{['fig:Q40x40']}: Mesh $\mathcal{T}_{h\ifstrempty{4}{}{_{4}}}^{Q}$. \ref{['fig:QD40x40']}: Mesh $\mathcal{T}_{h\ifstrempty{4}{}{_{4}}}^{DQ}$.
  • Figure 8: Test 2. Condition number of $\mathbf{K}$ vs. $k$. Left: $\epsilon = 1$. Right: $\epsilon = 10^{-6}$. First row: $\mathcal{T}_{h\ifstrempty{1}{}{_{1}}}^{Q}$. Second row: $\mathcal{T}_{h\ifstrempty{1}{}{_{1}}}^{DQ}$. Dirichlet BCs.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Remark 3.1