A family of stabilizer-free virtual elements on triangular meshes

TL;DR

This paper introduces a new family of stabilizer-free virtual elements on triangular meshes that maintain optimal convergence without the need for stabilizers, simplifying implementation while ensuring accuracy.

Contribution

The authors develop stabilizer-free $P_k$ virtual elements using a novel interpolation on macro-triangles, preserving polynomial properties and ensuring quasi-optimality.

Findings

Convergence at optimal order confirmed by numerical tests

Stabilizer can be dropped without losing accuracy

Method simplifies virtual element implementation

Abstract

A family of stabilizer-free $P_k$ virtual elements are constructed on triangular meshes. When choosing an accurate and proper interpolation, the stabilizer of the virtual elements can be dropped while the quasi-optimality is kept. The interpolating space here is the space of continuous $P_k$ polynomials on the Hsieh-Clough-Tocher macro-triangle, where the macro-triangle is defined by connecting three vertices of a triangle with its barycenter. We show that such an interpolation preserves $P_k$ polynomials locally and enforces the coerciveness of the resulting bilinear form. Consequently the stabilizer-free virtual element solutions converge at the optimal order. Numerical tests are provided to confirm the theory and to be compared with existing virtual elements.