Higher-order adaptive virtual element methods with contraction properties

TL;DR

This work extends adaptive virtual element methods to higher-order spaces ($k\ge2$) on triangular meshes for self-adjoint elliptic problems with Dirichlet data. It introduces a stabilization-free a posteriori framework that combines a residual estimator $\eta_{\mathcal{T}}$ with a virtual inconsistency estimator $\Psi_{\mathcal{T}}$, and proves convergence via a contraction analysis within an AVEM-like two-loop scheme. A key novelty is the use of the global index of hanging nodes and the concept of $\Lambda$-admissible meshes to control nonconformity and enable higher-order adaptivity without stabilization terms dominating. The paper also develops a detailed refinement analysis, quasi-orthogonality properties, and a GALERKIN module that delivers guaranteed error reduction, offering practical and theoretical tools for efficient higher-order VEM adaptivity.

Abstract

The realization of a standard Adaptive Finite Element Method (AFEM) preserves the mesh conformity by performing a completion step in the refinement loop: in addition to elements marked for refinement due to their contribution to the global error estimator, other elements are refined. In the new perspective opened by the introduction of Virtual Element Methods (VEM), elements with hanging nodes can be viewed as polygons with aligned edges, carrying virtual functions together with standard polynomial functions. The potential advantage is that all activated degrees of freedom are motivated by error reduction, not just by geometric reasons. This point of view is at the basis of the paper [L. Beirao da Veiga et al., Adaptive VEM: stabilization-free a posteriori error analysis and contraction property, SIAM Journal on Numerical Analysis, vol. 61, 2023], devoted to the convergence analysis of an adaptive VEM generated by the successive newest-vertex bisections of triangular elements without applying completion, in the lowest-order case (polynomial degree k=1). The purpose of this paper is to extend these results to the case of VEMs of order k>1 built on triangular meshes. The problem at hand is a variable-coefficient, second-order self-adjoint elliptic equation with Dirichlet boundary conditions; the data of the problem are assumed to be piecewise polynomials of degree k-1. By extending the concept of global index of a hanging node, under an admissibility assumption of the mesh, we derive a stabilization-free a posteriori error estimator. This is the sum of residual-type terms and certain virtual inconsistency terms (which vanish for k=1). We define an adaptive VEM of order k based on this estimator, and we prove its convergence by establishing a contraction result for a linear combination of (squared) energy norm of the error, residual estimator, and virtual inconsistency estimator.

Figures (5)

• Figure 1: Blue squares represent the $k+1$ equally-spaced nodes $\bm{\xi}_n$ on the edge $S$ before refinement. Red circles represent the $2k +1$ nodes that arise after refinement. We have denoted by $\bm{\zeta}_i$ the new nodes that do not coincide with any $\bm{\xi}_n$.
• Figure 2: Triangulation after the three refinements in the case $k=2$ (a) and in the case $k=3$ (b). Blue crosses represent the original degrees of freedom. Red squares, green circles and orange triangles are used for the degrees of freedom of the first, second and third refinement, respectively. All nodes are proper, except those on the horizontal line, whose global index is reported.
• Figure 3: The case $k=3$ with $3$ refinements of the edge $L$ (in blue) is shown. Red, green and orange lines are the lines needed to refine $L$ the first, the second and the third time respectively. Blue crosses are the degrees of freedom on $L$ of the function living on $E$. Red squares, green circles, orange diamonds are the degrees of freedom on $L$ generated after the first, the second and the third refinement of $L$.
• Figure 4: Blue square are the $k+1$ equi-spaced original nodes on the blue edge. Red points represent the nodes added after the refinement of the interval. Black lines show the shapes of the basis $\psi_i$, $i=1,\dots,k$, in the case $k=1$ (a), $k=2$ (b), $k=3$ (c).
• Figure 5: Black points are the proper nodes. Red points represent the hanging nodes generated after a refinement. In (a) the case $k=2$ is showed, $\bm{\zeta}_1$ is the hanging node obtained after the refinement of $\bm{\xi}_1$ and $\bm{\xi}_3$ and it is the midpoint of $\bm{\xi}_1$ and $\bm{\xi}_2$. We notice that if we have called the other red point $\bm{\zeta}_2$, $\bm{\xi}_1$ and $\bm{\xi}_3$ would have been switched. Analogusly, (b) represents the case $k=3$.

Theorems & Definitions (35)

• Lemma 3.1: Gakerkin quasi-orthogonality
• Definition 4.1: closest neighbors of a node
• Definition 4.2: global index of a node
• Proposition 5.1: scaled Poincaré inequality in $\mathbb{V}_\mathcal{T}$
• proof
• Remark 5.2
• Lemma 5.3: global interpolation error vs hierarchical errors
• Lemma 5.4
• proof
• Proposition 5.5: comparison between interpolation operators