Mixed virtual volume methods for elliptic problems

TL;DR

This paper introduces a new class of mixed virtual volume methods for elliptic problems on polygonal grids that produce symmetric, positive definite systems for the primary variable, simplifying velocity computation and ensuring conservation.

Contribution

The paper develops reduced mixed virtual volume methods that avoid Lagrangian multipliers and are equivalent to nonconforming virtual element methods, facilitating easier velocity computation.

Findings

Methods produce symmetric, positive definite systems.

Numerical results confirm optimal error estimates.

Velocity can be locally computed with simple formulas.

Abstract

We develop a class of mixed virtual volume methods for elliptic problems on polygonal/polyhedral grids. Unlike the mixed virtual element methods introduced in \cite{brezzi2014basic,da2016mixed}, our methods are reduced to symmetric, positive definite problems for the primary variable without using Lagrangian multipliers. We start from the usual way of changing the given equation into a mixed system using the Darcy's law, $\bu=-{\cal K} \nabla p$. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed virtual volume methods of all orders. We show that these new schemes are equivalent to the nonconforming virtual element methods for the primal variable $p$. Once the primary variable is computed solving the symmetric, positive definite system, all the degrees of freedom for the Darcy velocity are locally computed. Also, the $L^2$-projection onto the polynomial space is easy to compute. Hence our work opens an easy way to compute Darcy velocity on the polygonal/polyhedral grids. For the lowest order case, we give a formula to compute a Raviart-Thomas space like representation which satisfies the conservation law. An optimal error analysis is carried out and numerical results are presented which support the theory.