Some aspects on the computational implementation of diverse terms arising in mixed virtual element formulations

TL;DR

This paper details the computational implementation of integral terms in mixed virtual element methods for 2D pseudostress-velocity formulations, applied to Navier-Stokes equations, with numerical validation of convergence rates.

Contribution

It introduces a systematic way to implement integral terms for mixed-VEM with any polynomial degree and applies it to Navier-Stokes equations, including assembly algorithms and numerical testing.

Findings

Successful implementation for any polynomial degree k

Application to Navier-Stokes equations with Dirichlet boundary conditions

Numerical results confirm theoretical convergence rates

Abstract

In the present paper we describe the computational implementation of some integral terms that arise from mixed virtual element methods (mixed-VEM) in two-dimensional pseudostress-velocity formulations. The implementation presented here consider any polynomial degree $k \geq 0$ in a natural way by building several local matrices of small size through the matrix multiplication and the Kronecker product. In particular, we apply the foregoing mentioned matrices to the Navier-Stokes equations with Dirichlet boundary conditions, whose mixed-VEM formulation was originally proposed and analyzed in a recent work using virtual element subspaces for $H(\text{div})$ and $H^1$, simultaneously. In addition, an algorithm is proposed for the assembly of the associated global linear system for the Newton's iteration. Finally, we present a numerical example in order to illustrate the performance of the mixed-VEM scheme and confirming the expected theoretical convergence rates.