Stabilizer-free polygonal and polyhedral virtual elements

TL;DR

This paper introduces stabilizer-free virtual elements for polygonal and polyhedral meshes, achieving optimal convergence without stabilizers, validated through numerical examples in 2D and 3D.

Contribution

It develops a new class of stabilizer-free virtual elements using continuous polynomial spaces on subdivided meshes, ensuring positive-definiteness and optimal convergence.

Findings

Converge at optimal order in 2D and 3D.

Eliminate the need for stabilizers in virtual element methods.

Numerical examples validate theoretical convergence.

Abstract

Stabilizer-free $P_k$ virtual elements are constructed on polygonal and polyhedral meshes. Here the interpolating space is the space of continuous $P_k$ polynomials on a triangular-subdivision of each polygon, or a tetrahedral-subdivision of each polyhedron. With such an accurate and proper interpolation, the stabilizer of the virtual elements is eliminated while the system is kept positive-definite. We show that the stabilizer-free virtual elements converge at the optimal order in 2D and 3D. Numerical examples are computed, validating the theory.