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Nonconforming virtual element method for general second-order elliptic problems on curved domain

Yi Liu, Alessandro Russo

Abstract

This paper introduces a nonconforming virtual element method for general second-order elliptic problems with variable coefficients on domains with curved boundaries and curved internal interfaces. We prove arbitrary order optimal convergence in the energy and $L^2$ norms, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the method is shown to be comparable with the theoretical analysis.

Nonconforming virtual element method for general second-order elliptic problems on curved domain

Abstract

This paper introduces a nonconforming virtual element method for general second-order elliptic problems with variable coefficients on domains with curved boundaries and curved internal interfaces. We prove arbitrary order optimal convergence in the energy and norms, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the method is shown to be comparable with the theoretical analysis.

Paper Structure

This paper contains 22 sections, 32 theorems, 158 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The nonconforming virtual element method
  4. Mesh assumptions
  5. Broken spaces
  6. The nonconforming virtual element space
  7. The nonconforming virtual element method
  8. Polynomial and virtual element approximation estimates
  9. The difference between continuous and discrete bilinear forms
  10. The error between the bilinear functions $B(u_\pi,v_h)$ and $B_h(u_\pi,v_h)$
  11. The error between the bilinear functions $B(u_h,v_I)$ and $B_h(u_h,v_I)$
  12. The error between the bilinear functions $B(u_h,v_I)$ and $B_h(u_h,v_I)$ for stability analysis
  13. Stability analysis
  14. Convergence analysis
  15. An estimate in a discrete $H^1$-norm
  16. ...and 7 more sections

Key Result

Lemma 4.1

Given an element $K$ and $v_h$ in ${V}_h(K)$, we have

Figures (7)

  • Figure 9.1: Left-panel: an example of Voronoi mesh over $\Omega$. Right-panel: an example of concave mesh over $\Omega$.
  • Figure 9.2: Left-panel: the convergence of $E_{H^1}$. Right-panel: the convergence of $E_{L^2}$. The exact solution is $u_1$. We employ sequences of Voronoi meshes (first row) and concave meshes (second row) with decreasing mesh size are employed. The “orders” of the virtual element spaces are $k =1$, $2$, $3$, and $4$.
  • Figure 9.3: Domain $\Omega$ described in \ref{['eqn:sindomain']}.
  • Figure 9.4: Left-panel: an example of (curved) Voronoi mesh over $\Omega$. Right-panel: an example of (curved) quadrilateral mesh over $\Omega$.
  • Figure 9.5: Left-panels: the convergence of $E_{H^1}$. Right-panels: the convergence of $E_{L^2}$. The exact solution is $u_2$. We employ sequences of (curved) Voronoi meshes (first row) and (curved) square meshes (second row) with decreasing mesh size are employed. The "orders" of the virtual element spaces are $k=1$, $2$, $3$, and $4$.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Remark 2.1
  • Lemma 4.1
  • proof
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • proof
  • Lemma 4.6
  • Lemma 4.7
  • proof
  • ...and 43 more