Benchmarking stabilized and self-stabilized p-virtual element methods with variable coefficients

TL;DR

This paper conducts a comprehensive numerical comparison of stabilized and self-stabilized p-virtual element methods with variable coefficients, introducing a new projection operator that enhances robustness for high polynomial degrees.

Contribution

It provides the first detailed numerical analysis of stabilized and self-stabilized p-virtual element methods, introducing a new variable coefficient projection operator for improved robustness.

Findings

Self-stabilized and stabilization-free methods achieve optimal accuracy.

These methods have worse conditioning compared to stabilized methods.

The new projection operator improves robustness for large p values.

Abstract

Standard Virtual Element Methods (VEM) are based on polynomial projections and require a stabilization term to evaluate the contribution of the non-polynomial component of the discrete space. However, the stabilization term is not uniquely defined by the underlying variational formulation and is typically introduced in an ad hoc manner, potentially affecting the numerical response. Stabilization-free and self-stabilized formulations have been proposed to overcome this issue, although their theoretical analysis is still less mature. This paper provides an in-depth numerical investigation into different stabilized and self-stabilized formulations for the p-version of VEM. The results show that self-stabilized and stabilization-free formulations achieve optimal accuracy while suffering from worse conditioning. Moreover, a new projection operator, which explicitly accounts for variable coefficients, is introduced within the framework of standard virtual element spaces. Numerical results show that this new approach is more robust than the existing ones for large values of p.