# An H^1-conforming Virtual Element Method for Darcy equations and Brinkman equations

## TL;DR

This paper introduces an H^1-conforming Virtual Element Method for Darcy and Brinkman equations, achieving optimal convergence and stability across different flow regimes with rigorous analysis and numerical validation.

## Contribution

It develops a new Virtual Element space with computable projections and divergence-free properties, applicable to both Darcy and Brinkman equations, ensuring stability and optimal convergence.

## Key findings

- Optimal order convergence for Darcy equations
- Stable and robust method for Brinkman equations
- Numerical tests confirm theoretical error estimates

## Abstract

The focus of the present paper is on developing a Virtual Element Method for Darcy and Brinkman equations. In [15] we presented a family of Virtual Elements for Stokes equations and we defined a new Virtual Element space of velocities such that the associated discrete kernel is pointwise divergence-free. We use a slightly different Virtual Element space having two fundamental properties: the L^2-projection onto P_k is exactly computable on the basis of the degrees of freedom, and the associated discrete kernel is still pointwise divergence-free. The resulting numerical scheme for the Darcy equation has optimal order of convergence and H^1 conforming velocity solution. We can apply the same approach to develop a robust virtual element method for the Brinkman equation that is stable for both the Stokes and Darcy limit case. We provide a rigorous error analysis of the method and several numerical tests.