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3D Adaptive VEM with stabilization-free a posteriori error bounds

Stefano Berrone, Davide Fassino, Fabio Vicini

TL;DR

The paper advances Adaptive Virtual Element Methods to three dimensions by establishing a stabilization-free a posteriori error bound that bounds the stabilization term with a residual estimator, enabling energy-error control without stabilization terms. It defines Λ-admissible tetrahedral meshes with aligned faces/edges and develops the 3D VEM spaces, projections, and interpolants necessary for the analysis, culminating in a residual-based estimator η and a bound of the form |||u-u_T|||^2 ≤ C_L η^2 with a complementary bound involving the stabilization term. The GALERKIN module, together with Dörfler marking and NVB refinement, is shown to converge with estimator reduction and quasi-orthogonality, and numerical experiments on a Fichera corner demonstrate that AVEM can significantly reduce the number of 3D cells refined for a given tolerance compared to AFEM, while maintaining optimal convergence. The work thereby provides a practical, stabilization-free, 3D adaptivity framework for VEM that exploits hanging-node refinements to improve efficiency in complex geometries.

Abstract

The present paper extends the theory of Adaptive Virtual Element Methods (AVEMs) to the three-dimensional meshes showing the possibility to bound the stabilization term by the residual-type error estimator. This new bound enables a stabilization-free a posteriori control for the energy error. Following the recent studies for the bi-dimensional case, we investigate the case of tetrahedral elements with aligned edges and faces. We believe that the AVEMs can be an efficient strategy to address the mesh conforming requirements of standard three-dimensional Adaptive Finite Element Methods (AFEMs), which typically extend the refinement procedure to non-marked mesh cells. Indeed, numerical tests on the Fichera corner shape domain show that this method can reduce the number of three-dimensional cells generated in the refinement process by about 30% with compared to standard AFEMs, for a given error threshold.

3D Adaptive VEM with stabilization-free a posteriori error bounds

TL;DR

The paper advances Adaptive Virtual Element Methods to three dimensions by establishing a stabilization-free a posteriori error bound that bounds the stabilization term with a residual estimator, enabling energy-error control without stabilization terms. It defines Λ-admissible tetrahedral meshes with aligned faces/edges and develops the 3D VEM spaces, projections, and interpolants necessary for the analysis, culminating in a residual-based estimator η and a bound of the form |||u-u_T|||^2 ≤ C_L η^2 with a complementary bound involving the stabilization term. The GALERKIN module, together with Dörfler marking and NVB refinement, is shown to converge with estimator reduction and quasi-orthogonality, and numerical experiments on a Fichera corner demonstrate that AVEM can significantly reduce the number of 3D cells refined for a given tolerance compared to AFEM, while maintaining optimal convergence. The work thereby provides a practical, stabilization-free, 3D adaptivity framework for VEM that exploits hanging-node refinements to improve efficiency in complex geometries.

Abstract

The present paper extends the theory of Adaptive Virtual Element Methods (AVEMs) to the three-dimensional meshes showing the possibility to bound the stabilization term by the residual-type error estimator. This new bound enables a stabilization-free a posteriori control for the energy error. Following the recent studies for the bi-dimensional case, we investigate the case of tetrahedral elements with aligned edges and faces. We believe that the AVEMs can be an efficient strategy to address the mesh conforming requirements of standard three-dimensional Adaptive Finite Element Methods (AFEMs), which typically extend the refinement procedure to non-marked mesh cells. Indeed, numerical tests on the Fichera corner shape domain show that this method can reduce the number of three-dimensional cells generated in the refinement process by about 30% with compared to standard AFEMs, for a given error threshold.
Paper Structure (10 sections, 18 theorems, 104 equations, 7 figures)

This paper contains 10 sections, 18 theorems, 104 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The problem and the discretization
  3. VEM spaces and projectors
  4. The discrete problem
  5. The preliminary properties
  6. A posteriori error analysis
  7. The module GALERKIN
  8. The convergence of GALERKIN
  9. Numerical experiments
  10. Conclusions

Key Result

Lemma 2.6

For the projection operator $\Pi^\nabla_E$ it holds the following where the hidden constant does not depend on the diameter $h_E$. Furthermore,

Figures (7)

  • Figure 1: Three successive refinements of a tetrahedron. Blue lines represent the edges added at each refinement. Black points and red points represent the proper and the hanging nodes, respectively. The numbers are the global indices.
  • Figure 2: Element $E\in \mathcal{T}$ obtained after the refinement of element $T$ via the NVB, where the edge bisected is the one with endpoints $\bm{x}'_0$ and $\bm{x}"_0$. $\lambda_j,\lambda_k,\lambda_\ell,\lambda_m$ are the global indices of the vertices of the element $T$, whereas $\lambda_i,\lambda_j,\lambda_k,\lambda_m$ are the global indices of the vertices of $E$.
  • Figure 3: The four elements of the chain \ref{['eq:chain']} from the light blue tetrahedral-shape element $E$ to the element with a proper node $T_3$. This chain can also be seen as three successive refinements of element $T_3$. Blue lines represent the edges added at each refinement. Black points represent the vertices of the considered elements, red points denote the hanging nodes. The numbers next to the vertices are their global indices. The fixed maximum global index is $\Lambda =2$.
  • Figure 4: Black and red points represent the proper and the hanging nodes respectively. The light blue element has only hanging nodes as its vertices. Notice that this is obtained by a further refinement of the last element of Figure \ref{['Figure:GlobalIndex']}.
  • Figure 5: Two elements $E$ (peach) and $E'$ (light blue) sharing an edge and the node $\bm{\hat{x}}$ (red) that violates Assumption \ref{['ass:LambdaMax']} (a). If $\bm{\hat{x}}$ does belong to the newest-edge, one refinement of $E$ restores the $\Lambda$-admissibility (b). If $\bm{\hat{x}}$ does not belong to the newest-edge, two and three refinements of $E$ are needed (c) and (d), respectively. Purple, green and yellow are the new faces separating the tetrahedral-shape elements. In (b), (c), and (d) $E'$ has not been represented for graphic purposes.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 2.1: global index of a node
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7: Galerkin quasi-orthogonality
  • proof
  • Lemma 2.8
  • proof
  • ...and 24 more