Nitsche stabilized Virtual element approximations for a Brinkman problem with mixed boundary conditions

TL;DR

This paper develops and analyzes a Nitsche stabilized virtual element method for the Brinkman problem with mixed boundary conditions, providing stability, error estimates, and numerical validation.

Contribution

It introduces a novel Nitsche stabilized virtual element scheme for the Brinkman problem with mixed boundary conditions, including slip conditions, with rigorous analysis and numerical tests.

Findings

Stable and robust scheme independent of viscosity

Optimal a priori error estimates established

Numerical tests confirm theoretical convergence and robustness

Abstract

In this paper, we formulate, analyse and implement the discrete formulation of the Brinkman problem with mixed boundary conditions, including slip boundary condition, using the Nitsche's technique for virtual element methods. The divergence conforming virtual element spaces for the velocity function and piecewise polynomials for pressure are approached for the discrete scheme. We derive a robust stability analysis of the Nitsche stabilized discrete scheme for this model problem. We establish an optimal and vigorous a priori error estimates of the discrete scheme with constants independent of the viscosity. Moreover, a set of numerical tests demonstrates the robustness with respect to the physical parameters and verifies the derived convergence results.