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Isoparametric Virtual Element Methods

Andrea Cangiani, Andreas Dedner, Matthew Hubbard, Harry Wells

TL;DR

The paper develops two isoparametric Virtual Element Methods (IsoVEMs) for 2D linear elliptic problems on curved domains by either mapping the problem to a reference domain or by constructing a computable virtual domain through a projected map. It establishes stability and optimal convergence in the $H^1$-norm and $L^2$-norm, deriving detailed Jacobian and projection error analyses and Strang-type bounds. Numerical experiments in the DUNE framework validate the theory, showing optimal rates and illustrating the influence of the mapping degree on accuracy. The work provides a robust, computable pathway for high-order VEMs on curved geometries, with clear potential for extensions to time-dependent and higher-dimensional problems.

Abstract

We present two approaches to constructing isoparametric Virtual Element Methods of arbitrary order for linear elliptic partial differential equations on general two-dimensional domains. The first method approximates the variational problem transformed onto a computational reference domain. The second method computes a virtual domain and uses bespoke polynomial approximation operators to construct a computable method. Both methods are shown to converge optimally, a behaviour confirmed in practice for the solution of problems posed on curved domains.

Isoparametric Virtual Element Methods

TL;DR

The paper develops two isoparametric Virtual Element Methods (IsoVEMs) for 2D linear elliptic problems on curved domains by either mapping the problem to a reference domain or by constructing a computable virtual domain through a projected map. It establishes stability and optimal convergence in the -norm and -norm, deriving detailed Jacobian and projection error analyses and Strang-type bounds. Numerical experiments in the DUNE framework validate the theory, showing optimal rates and illustrating the influence of the mapping degree on accuracy. The work provides a robust, computable pathway for high-order VEMs on curved geometries, with clear potential for extensions to time-dependent and higher-dimensional problems.

Abstract

We present two approaches to constructing isoparametric Virtual Element Methods of arbitrary order for linear elliptic partial differential equations on general two-dimensional domains. The first method approximates the variational problem transformed onto a computational reference domain. The second method computes a virtual domain and uses bespoke polynomial approximation operators to construct a computable method. Both methods are shown to converge optimally, a behaviour confirmed in practice for the solution of problems posed on curved domains.
Paper Structure (28 sections, 22 theorems, 162 equations, 4 figures)

This paper contains 28 sections, 22 theorems, 162 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model problem
  4. Reference domain
  5. $C^0$-conforming VEM
  6. Enhanced curved VEM spaces
  7. Curved VEM
  8. Virtual element mapping
  9. Formulation of the isoparametric VEMs
  10. Reference IsoVEM formulation
  11. Physical IsoVEM formulation
  12. Reference IsoVEM analysis
  13. Jacobian error & stability estimates
  14. Well-posedness
  15. Reference IsoVEM error estimate
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\omega \subset \mathbb{R}^2$ be bounded and Lipschitz. For all $p \in [1,\infty]$ and integers $s \geq 0$ there exists an extension operator $\mathfrak{E} : W_p^s(\omega) \rightarrow W_p^s(\mathbb{R}^2)$ such that, for all $v \in W^s_p(\omega)$, $\mathfrak{E} v = v$ a.e. in $\omega$ and $\left\

Figures (4)

  • Figure 1: From left to right, a physical element $E$ and its virtual interpolating polygons $E_h$ obtained with virtual map of order $l=1$, $l=2$, and $l=3$.
  • Figure 2: The annulus (left) and plane (right) domains approximated by a VEM mapping on a discretisation of $\hat{\Omega} = [0,1]^2$ using 800 polygonal elements.
  • Figure 3: Error plots for the annulus mapping problem with isoparametric elements of degree $l,k=1,2,3$.
  • Figure 4: Error plots for the plane mapping problem with isoparametric elements ($l=k$) of degree $k=1,2,3$.

Theorems & Definitions (51)

  • Remark 1
  • Remark 2
  • Theorem 1: The Stein Extension Theorem
  • Theorem 2: $L^2$-projection accuracy
  • Remark 3
  • Definition 1: Virtual element $\text{DoF}$
  • Remark 4
  • Theorem 3: VEM interpolation error estimate
  • Remark 5
  • Theorem 4: Strang-type bound
  • ...and 41 more