A priori and a posteriori error estimates of a really pressure-robust virtual element method for the incompressible Brinkman problem

TL;DR

This paper develops and analyzes a pressure-robust virtual element method for the incompressible Brinkman problem, providing optimal error estimates, adaptive refinement strategies, and demonstrating robustness and accuracy through numerical tests.

Contribution

It introduces a divergence-preserving reconstruction operator and a residual-based a posteriori error estimator for the virtual element method applied to the Brinkman problem.

Findings

Velocity error is independent of pressure and viscosity.

The a posteriori estimator provides reliable error bounds.

Numerical experiments confirm robustness and efficiency.

Abstract

This paper presents both a priori and a posteriori error analyses for a really pressure-robust virtual element method to approximate the incompressible Brinkman problem. We construct a divergence-preserving reconstruction operator using the Raviart-Thomas element for the discretization on the right-hand side. The optimal priori error estimates are carried out, which imply the velocity error in the energy norm is independent of both the continuous pressure and the viscosity. Taking advantage of the virtual element method's ability to handle more general polygonal meshes, we implement effective mesh refinement strategies and develop a residual-type a posteriori error estimator. This estimator is proven to provide global upper and local lower bounds for the discretization error. Finally, some numerical experiments demonstrate the robustness, accuracy, reliability and efficiency of the method.