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Effective polygonal mesh generation and refinement for VEM

Stefano Berrone, Fabio Vicini

Abstract

In the present work we introduce a novel refinement algorithm for two-dimensional elliptic partial differential equations discretized with Virtual Element Method (VEM). The algorithm improves the numerical solution accuracy and the mesh quality through a controlled refinement strategy applied to the generic polygonal elements of the domain tessellation. The numerical results show that the outlined strategy proves to be versatile and applicable to any two-dimensional problem where polygonal meshes offer advantages. In particular, we focus on the simulation of flow in fractured media, specifically using the Discrete Fracture Network (DFN) model. A residual a-posteriori error estimator tailored for the DFN case is employed. We chose this particular application to emphasize the effectiveness of the algorithm in handling complex geometries. All the numerical tests demonstrate optimal convergence rates for all the tested VEM orders.

Effective polygonal mesh generation and refinement for VEM

Abstract

In the present work we introduce a novel refinement algorithm for two-dimensional elliptic partial differential equations discretized with Virtual Element Method (VEM). The algorithm improves the numerical solution accuracy and the mesh quality through a controlled refinement strategy applied to the generic polygonal elements of the domain tessellation. The numerical results show that the outlined strategy proves to be versatile and applicable to any two-dimensional problem where polygonal meshes offer advantages. In particular, we focus on the simulation of flow in fractured media, specifically using the Discrete Fracture Network (DFN) model. A residual a-posteriori error estimator tailored for the DFN case is employed. We chose this particular application to emphasize the effectiveness of the algorithm in handling complex geometries. All the numerical tests demonstrate optimal convergence rates for all the tested VEM orders.
Paper Structure (14 sections, 20 equations, 17 figures, 1 table, 4 algorithms)

This paper contains 14 sections, 20 equations, 17 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Setting and Target
  3. VEM Space
  4. Target
  5. The REFINE module
  6. Contributions of the new algorithm
  7. The problem and the discretization
  8. DFN Discretization
  9. DFN Mesh Adaptivity
  10. Numerical Results
  11. Test 1: Classic bi-dimensional Domain
  12. Test 2: Regular DFN with three Fractures
  13. Test 3: Random DFN
  14. Conclusions

Figures (17)

  • Figure 1: Refinement of $E$ - Case $1$ - $c_{\rho} = 0.5$ and $\frac{4}{5} < p < 1$
  • Figure 2: Refinement of $E$ - Case $2$ - $c_{\rho} = 0.5$
  • Figure 3: Refinement of $E$ - Measure of the local error estimator on a numerical case
  • Figure 4: Test $1$ - Domain with solution and, $k = 1$, $c_{al} = 1.0$.
  • Figure 5: Test $1$ - mesh statistics, $k=1$, $c_{al} = 1.0$.
  • ...and 12 more figures

Theorems & Definitions (1)

  • Remark : Local flux estimation