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Generalised gradients for virtual elements and applications to a posteriori error analysis

Théophile Chaumont-Frelet, Joscha Gedicke, Lorenzo Mascotto

Abstract

We rewrite the standard nodal virtual element method as a generalised gradient method. This re-formulation allows for computing a reliable and efficient error estimator by locally reconstructing broken fluxes and potentials on elemental subtriangulations. We prove the usual upper and lower bounds with constants independent of the stabilisation of the method and, under technical assumptions on the mesh, the degree of accuracy.

Generalised gradients for virtual elements and applications to a posteriori error analysis

Abstract

We rewrite the standard nodal virtual element method as a generalised gradient method. This re-formulation allows for computing a reliable and efficient error estimator by locally reconstructing broken fluxes and potentials on elemental subtriangulations. We prove the usual upper and lower bounds with constants independent of the stabilisation of the method and, under technical assumptions on the mesh, the degree of accuracy.
Paper Structure (31 sections, 14 theorems, 109 equations, 5 figures, 2 tables)

This paper contains 31 sections, 14 theorems, 109 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. Functional spaces
  4. The continuous problem
  5. Regular polygonal meshes
  6. Standard polynomial spaces
  7. The virtual element method and generalised gradients
  8. Virtual element spaces
  9. Stabilisations and polynomial projectors
  10. The standard formulation of the VEM
  11. The VEM as a generalised gradient method
  12. Technical results
  13. Virtual element partition of unity
  14. Discrete stable minimisation
  15. A posteriori error estimator
  16. ...and 16 more sections

Key Result

Lemma 3.2

For all $\phi_h$ in $V_h(K)$, there exist unique $\mu_h^K(\phi_h)$ in $\mathbb P_{p}(\mathcal{E}^K)$ and $r_h^K(\phi_h)$ in $\mathbb P_{p-2}(K)$ such that

Figures (5)

  • Figure 1: Examples of configurations allowed (left panel) and forbidden (right panel) under Assumption \ref{['assumption_faces']} for the derivation of $p$-robust estimates. The $h$-version allows for both configurations.
  • Figure 2: Effectivity index $\mathcal{I}$ in \ref{['effectivity-index']} versus the number of degrees of freedom $\mathcal{N}_\ell$ for the test cases $u_1$ (left panel) and $u_2$ (right panel) under uniform and adaptive $h$-refinements. We employ an initial uniform Cartesian mesh of 4 or 12 elements, respectively, and $p=1$ and $2$.
  • Figure 3: Effectivity index $\mathcal{I}$ in \ref{['effectivity-index']} for the test case $u_2$ under uniform $p$-refinements.
  • Figure 4: Enumeration in an interior vertex patch
  • Figure 5: Symmetrisation of boundary patch

Theorems & Definitions (32)

  • Remark 2.2
  • Remark 3.1
  • Lemma 3.2: Stabilisation lifting
  • proof
  • Definition 3.3: Generalised gradient
  • Theorem 3.4: Rewriting of the VEM
  • proof
  • Remark 4.1
  • Lemma 4.2
  • proof
  • ...and 22 more