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Stability and interpolation estimates of Hellinger-Reissner virtual element spaces

Michele Botti, Lorenzo Mascotto, Giuseppe Vacca, Michele Visinoni

TL;DR

This work develops a rigorous stability and interpolation theory for Hellinger--Reissner virtual element spaces in 3D linear elasticity on polytopic meshes. It introduces two explicit stabilizations (a projection-based one and a dofi-dofi one) that yield stability results with constants depending only on the mesh aspect ratio and the polynomial degree $p$, and derives interpolation estimates with explicit constants and commutativity properties for the divergence. A key contribution is the demonstration of these properties for HR-VE spaces with fixed $p$ in three dimensions, together with an extensive numerical study of stability constants on badly shaped elements and under increasing accuracy. The numerical experiments reveal robustness of the discretization under challenging geometries and higher $p$, while also highlighting limitations that motivate future stabilization strategies for degenerate geometries.

Abstract

We prove stability and interpolation estimates for Hellinger-Reissner virtual elements; the constants appearing in such estimates only depend on the aspect ratio of the polytope under consideration and the degree of accuracy of the scheme. We further investigate numerically the behaviour of the constants appearing in the stability estimates on sequences of badly-shaped polytopes and for increasing degree of accuracy.

Stability and interpolation estimates of Hellinger-Reissner virtual element spaces

TL;DR

This work develops a rigorous stability and interpolation theory for Hellinger--Reissner virtual element spaces in 3D linear elasticity on polytopic meshes. It introduces two explicit stabilizations (a projection-based one and a dofi-dofi one) that yield stability results with constants depending only on the mesh aspect ratio and the polynomial degree , and derives interpolation estimates with explicit constants and commutativity properties for the divergence. A key contribution is the demonstration of these properties for HR-VE spaces with fixed in three dimensions, together with an extensive numerical study of stability constants on badly shaped elements and under increasing accuracy. The numerical experiments reveal robustness of the discretization under challenging geometries and higher , while also highlighting limitations that motivate future stabilization strategies for degenerate geometries.

Abstract

We prove stability and interpolation estimates for Hellinger-Reissner virtual elements; the constants appearing in such estimates only depend on the aspect ratio of the polytope under consideration and the degree of accuracy of the scheme. We further investigate numerically the behaviour of the constants appearing in the stability estimates on sequences of badly-shaped polytopes and for increasing degree of accuracy.

Paper Structure

This paper contains 12 sections, 13 theorems, 111 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Hellinger--Reissner virtual element spaces and forms
  3. Explicit stabilizations and stability bounds
  4. A "projection-based" stabilization
  5. A "dofi-dofi" stabilization
  6. Interpolation estimates
  7. A numerical investigation on the stability constants
  8. Computation of the coercivity and continuity constants
  9. Stability constants on sequences of badly-shaped elements
  10. Stability constants increasing the degree of accuracy
  11. Conclusions
  12. Well-posedness of the linear elasticity problem in mixed formulation with essential boundary conditions

Key Result

Proposition 3.1

The bilinear form $S^K(\cdot,\cdot)$ defined in stabilization:general-order is computable by means of the degrees of freedom face-moments and interior-moments, and satisfies the stability bounds stability-bounds. In fact, the result holds true also for tensors $\underline{\boldsymbol \tau}_h$ such t

Figures (3)

  • Figure 1: A sequence of 2D hourglass shaped elements for a compressible material.
  • Figure 2: A sequence of 3D hourglass shaped elements for a compressible material.
  • Figure 3: A sequence of 2D trapezoidal elements for an incompressible material.

Theorems & Definitions (29)

  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Remark 4
  • Remark 5
  • ...and 19 more