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Conforming virtual element method for nondivergence form linear elliptic equations with Cordes coefficients

Guillaume Bonnet, Andrea Cangiani, Ricardo H. Nochetto

Abstract

We propose and analyze an $H^2$-conforming Virtual Element Method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension $d \ge 2$. The analysis relies on the continuous Miranda-Talenti estimate for convex domains $Ω$ and is rather elementary. We prove stability and error estimates in $H^2(Ω)$, including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in $H^2(Ω)$.

Conforming virtual element method for nondivergence form linear elliptic equations with Cordes coefficients

Abstract

We propose and analyze an -conforming Virtual Element Method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension . The analysis relies on the continuous Miranda-Talenti estimate for convex domains and is rather elementary. We prove stability and error estimates in , including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in .
Paper Structure (22 sections, 31 theorems, 97 equations, 6 figures)

This paper contains 22 sections, 31 theorems, 97 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The continuous problem
  3. The Cordes condition
  4. Variational formulation and well-posedness
  5. Virtual element discretization
  6. Description of the scheme
  7. Existence of a unique discrete solution
  8. Error estimate
  9. The effect of numerical integration
  10. Realization of the virtual element framework
  11. Notation
  12. Local virtual space in two-dimensions
  13. Local virtual space in three-dimensions
  14. Local projection operator and stabilization term
  15. Global spaces and quasi-interpolation operators
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $0 \leq \mu < 1$, and let $\mathbf{A} \in \mathbb{R}^{d \times d}$ be symmetric positive definite. The following are equivalent:

Figures (6)

  • Figure 1: Degrees of freedom of the $H^2$-conforming VEM for $m=2,3,4$.
  • Figure 2: Degrees of freedom of the $H^2$-conforming VEM for $m=2,3,4$.
  • Figure 3: Illustration of a construction satisfying the condition \ref{['eq:interp_boundary_dependence']} in dimension two. Left: on vertices of $\Omega$, the value degree of freedom and both components of the gradient degrees of freedom need to be imposed by the boundary data. Right: on vertices of $\mathcal{T}_h$ that belong to the interior of an edge of $\Omega$, only the value degree of freedom and the tangential part of the gradient degrees of freedom (in blue, as opposed to the normal part in green) need to be imposed by the boundary data.
  • Figure 4: First four randomly generated polygonal meshes, in dimension two.
  • Figure 6: Example 2. Top: dimension two, bottom: dimension three. As expected, the optimal rate of convergence is observed in the $H^2$ norm regardless of the mesh. This highlights the fact that the discontinuity of $A$ and $f$ does not have any negative effect on the rate of convergence.
  • ...and 1 more figures

Theorems & Definitions (72)

  • Definition 2.1: Cordes condition
  • Remark 2.1
  • Proposition 2.1: characterizations of the Cordes condition
  • proof
  • Definition 2.2: admissible scaling
  • Proposition 2.2
  • proof
  • Proposition 2.3: consistency
  • proof
  • Proposition 2.4: continuity
  • ...and 62 more