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Nodal auxiliary space preconditioners for mixed virtual element methods

Wietse Boon, Erik Nilsson

TL;DR

Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes.

Abstract

We propose nodal auxiliary space preconditioners for facet and edge virtual elements of lowest order by deriving discrete regular decompositions on polytopal grids and generalizing the Hiptmair-Xu preconditioner to the virtual element framework. The preconditioner consists of solving a sequence of elliptic problems on the nodal virtual element space, combined with appropriate smoother steps. Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes. Moreover, we observe numerically that the preconditioner is robust on meshes containing elements with high aspect ratios.

Nodal auxiliary space preconditioners for mixed virtual element methods

TL;DR

Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes.

Abstract

We propose nodal auxiliary space preconditioners for facet and edge virtual elements of lowest order by deriving discrete regular decompositions on polytopal grids and generalizing the Hiptmair-Xu preconditioner to the virtual element framework. The preconditioner consists of solving a sequence of elliptic problems on the nodal virtual element space, combined with appropriate smoother steps. Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes. Moreover, we observe numerically that the preconditioner is robust on meshes containing elements with high aspect ratios.
Paper Structure (32 sections, 24 theorems, 86 equations, 3 figures, 5 tables)

This paper contains 32 sections, 24 theorems, 86 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Continuous function spaces
  4. Model problems
  5. Exact co-chain complexes
  6. Regular decomposition
  7. Virtual element spaces of lowest order
  8. Useful inequalities for virtual element spaces
  9. The auxiliary nodal space
  10. A space of regular potentials
  11. Regular decomposition for virtual elements
  12. A discrete co-chain complex
  13. Stable co-chain projections
  14. Exactness of the discrete complex
  15. Approximation properties for regular functions
  16. ...and 17 more sections

Key Result

Lemma 2.1

If a co-chain complex $(V^\bullet, d)$ is exact, then:

Figures (3)

  • Figure 1: A discrete co-chain complex formed by the nodal, edge, facet, and cell virtual element spaces $V_h^k$ on a regular dodecahedron.
  • Figure 2: The two coarsest meshes (with $N=4,8$ respectively) used in the mesh size dependency experiments of \ref{['sub: Hdiv projection problem', 'sub: Darcy problem']}. This sequence of meshes satisfies \ref{['ass:mesh']}.
  • Figure 3: Meshes used in the numerical experiments for the aspect ratio tests. The elements crossed by the red line at $y = 0.5+\epsilon$ are split in two by introducing new nodes at the intersections with mesh edges.

Theorems & Definitions (48)

  • Remark 2.1: The two-dimensional $\mathop{\mathrm{curl}}\nolimits$
  • Lemma 2.1: Poincaré inequality ArnFEEC18
  • Lemma 2.2: Stable potentials
  • proof
  • Lemma 2.3: Continuous regular decomposition HiptXu07
  • Lemma 2.4: Local $H^1$-regularity
  • proof
  • Lemma 2.5: Nodal inverse inequalities Huang2018Errors2D, Huang2023Estimates
  • Lemma 2.6: Nodal trace inequalities
  • proof
  • ...and 38 more