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

TL;DR

This work develops a Nitsche stabilized divergence‑conforming virtual element method for the Brinkman problem with mixed boundary conditions on polygonal meshes. By constructing appropriate VEM spaces and projection operators, and by incorporating Nitsche terms to weakly enforce Dirichlet and slip conditions, the authors prove stability via a discrete inf‑sup condition and derive parameter‑robust, optimal a priori error estimates independent of the viscosity. The methodology is validated through extensive numerical experiments on diverse mesh families, showing optimal convergence rates and robustness to $\nu$ and complex boundary configurations. The results demonstrate that the proposed Nitsche VEM provides a flexible and reliable framework for simulating Brinkman flows in porous media with heterogeneous boundary conditions and polygonal meshes, with potential for high‑order accuracy and practical engineering applications.

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.

Figures (11)

• Figure 1: Sample geometry of the considered Brinkman domain with mixed boundary conditions.
• Figure 1: Test \ref{['subsec:squaredomain']}. Error curves of the virtual element scheme for the Brinkman equations using different meshes. Here, we set the parameters $\mathbb{K}=\mathbb{I}$, and $\nu=1$.
• Figure 2: Test \ref{['subsec:squaredomain']}. Scalar components of the computed velocity $\boldsymbol{u}_{i,h}, i=1,2$ in different meshes together with the corresponding field $\boldsymbol{u}_h$.
• Figure 3: Test \ref{['subsec:squaredomain']}. Comparison between the computed pressure $p_h$ on different polygonal meshes.
• Figure 4: Test \ref{['subsec:flow_past_cylinder']}. Computed scalar components of the velocity field (top) along with the corresponding velocity field and streamlines (bottom) in the cross-flow model.

Theorems & Definitions (16)

• lemma thmcounterlemma
• remark thmcounterremark
• remark thmcounterremark
• remark thmcounterremark
• lemma thmcounterlemma: Discrete trace inequality
• lemma thmcounterlemma: Discrete inverse inequality
• lemma thmcounterlemma
• lemma thmcounterlemma
• lemma thmcounterlemma
• lemma thmcounterlemma