Orthogonal polynomial bases in the Mixed Virtual Element Method

TL;DR

This paper introduces a specialized orthogonal vector-polynomial basis for the mixed Virtual Element Method, improving numerical stability and solution quality, especially in complex geometries like Discrete Fracture Networks.

Contribution

It develops a new orthogonal basis tailored for the mixed formulation of VEM, addressing ill-conditioning issues not solved by primal basis extensions.

Findings

Enhanced numerical stability in mixed VEM formulations.

High-quality solutions in complex geometries like DFN.

Effective handling of badly-shaped elements.

Abstract

The use of orthonormal polynomial bases has been found to be efficient in preventing ill-conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly-shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill-conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector-polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high-quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly-shaped elements, is proposed to validate our procedures.