A lowest order stabilization-free mixed Virtual Element Method

TL;DR

This paper introduces a stabilization-free mixed Virtual Element Method for the Laplacian problem, reducing computational costs while ensuring stability and convergence, supported by theoretical analysis and numerical tests.

Contribution

It presents the first stabilization-free Virtual Element Method for the Laplacian in mixed form, using a projection on harmonic polynomial gradients to enhance efficiency.

Findings

The method is stable and convergent on quadrilateral meshes.

Numerical tests confirm the scheme's effectiveness.

Computational costs are reduced compared to traditional methods.

Abstract

We initiate the design and the analysis of stabilization-free Virtual Element Methods for the laplacian problem written in mixed form. A Virtual Element version of the lowest order Raviart-Thomas Finite Element is considered. To reduce the computational costs, a suitable projection on the gradients of harmonic polynomials is employed. A complete theoretical analysis of stability and convergence is developed in the case of quadrilateral meshes. Some numerical tests highlighting the actual behaviour of the scheme are also provided.