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The virtual element method on polygonal pixel-based tessellations

Silvia Bertoluzza, Monica Montardini, Micol Pennacchio, Daniele Prada

TL;DR

This work develops and validates a boundary-corrected virtual element method (VEM) for Poisson problems on curved domains approximated by polygonal, image-derived tessellations. By discretizing the approximated domain with polygonal agglomerations and employing Nitsche’s method with SBM/BDT-style corrections, the authors restore high-order accuracy even when the boundary does not align with the mesh. A key contribution is a static condensation strategy that eliminates lazy degrees of freedom tied to many small edges, maintaining efficiency for high-order approximations. Numerical results on disk and bean domains, including elasticity, demonstrate stability and optimal convergence across a range of orders, highlighting the method’s potential for image-based geometry in two dimensions and its adaptability to three dimensions in future work.

Abstract

We analyze and validate the virtual element method combined with a boundary correction similar to the one in [1,2], to solve problems on two dimensional domains with curved boundaries approximated by polygonal domains. We focus on the case of approximating domains obtained as the union of squared elements out of a uniform structured mesh, such as the one that naturally arises when the domain is issued from an image. We show, both theoretically and numerically, that resorting to polygonal elements allows the assumptions required for stability to be satisfied for any polynomial order. This allows us to fully exploit the potential of higher order methods. Efficiency is ensured by a novel static condensation strategy acting on the edges of the decomposition.

The virtual element method on polygonal pixel-based tessellations

TL;DR

This work develops and validates a boundary-corrected virtual element method (VEM) for Poisson problems on curved domains approximated by polygonal, image-derived tessellations. By discretizing the approximated domain with polygonal agglomerations and employing Nitsche’s method with SBM/BDT-style corrections, the authors restore high-order accuracy even when the boundary does not align with the mesh. A key contribution is a static condensation strategy that eliminates lazy degrees of freedom tied to many small edges, maintaining efficiency for high-order approximations. Numerical results on disk and bean domains, including elasticity, demonstrate stability and optimal convergence across a range of orders, highlighting the method’s potential for image-based geometry in two dimensions and its adaptability to three dimensions in future work.

Abstract

We analyze and validate the virtual element method combined with a boundary correction similar to the one in [1,2], to solve problems on two dimensional domains with curved boundaries approximated by polygonal domains. We focus on the case of approximating domains obtained as the union of squared elements out of a uniform structured mesh, such as the one that naturally arises when the domain is issued from an image. We show, both theoretically and numerically, that resorting to polygonal elements allows the assumptions required for stability to be satisfied for any polynomial order. This allows us to fully exploit the potential of higher order methods. Efficiency is ensured by a novel static condensation strategy acting on the edges of the decomposition.
Paper Structure (18 sections, 3 theorems, 84 equations, 13 figures, 2 tables)

This paper contains 18 sections, 3 theorems, 84 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Boundary correction methods in the finite element framework
  3. The Virtual Element Discretization
  4. The tessellation
  5. The virtual element space
  6. Nitsche's method with boundary correction for VEM
  7. Static condensation of the "lazy" degrees of freedom
  8. Numerical results
  9. Test 1 -- Disk domain
  10. Effect of the refinement parameter $\widehat{\tau}$
  11. Effect of the choice of the stabilization parameter $\gamma$
  12. Robustness with respect to the parameter $\widehat{\tau}$
  13. Test 2 -- Bean domain
  14. Application to elasticity
  15. Conclusions and perspectives
  16. ...and 3 more sections

Key Result

Lemma 3.4

Under Assumption shape_regular, for all $\varphi \in H^1(K)$ it holds that Additionally, for all $p \in \mathbb{P}_{k}$ we have that Moreover, provided Assumption additional also holds, for all $v \in H^{r}(K)$, $r \geq 1$, we have that where $\mathcal{E}^K$ denotes the set of edges of the polygon $K$.

Figures (13)

  • Figure 1: Three possible elements of the tessellation $\mathcal{T}_H$. For the sake of the exposition, boundary edges (with vertices marked in red) are never agglomerated to form larger edges, even when this is possible. Agglomeration of interior edges (vertices in blue) into larger edges fits instead in our exposition.
  • Figure 2: Three examples of the auxiliary triangulation $\widetilde{\mathcal{T}}_K$. As one can see in the leftmost example, the presence of two adjacent boundary edges with very different length results in a badly shaped triangle. Adding few nodes, as in the central example, may improve the shape regularity of the triangulation. As the $\widetilde{\mathcal{T}}_K$ is allowed to have a number of elements as large as needed, the presence of a large number of very small edges does not, in itself, result in badly shaped triangulation (see the rightmost example).
  • Figure 3: An approximate domains $\Omega_h$ falling in our framework. The theoretical framework does not in principle require the extrapolation direction $\sigma$ to coincide with either the normal $\nu_h$ to the approximate boundary or the normal $\nu^\star$ to the physical boundary, though the latter is generally the best choice.
  • Figure 4: One element with 27 edges and 5 macro edges.
  • Figure 5: The solution to test problem 2 (top) and one of the meshes used in the tests. Remark that the approximate domain $\Omega_h$ is included in $\Omega$.
  • ...and 8 more figures

Theorems & Definitions (11)

  • Remark 2.1
  • Remark 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Remark 3.7
  • Remark 3.8
  • Remark 3.9
  • Remark 4.1
  • Remark 4.2
  • ...and 1 more