A two-level semi-hybrid-mixed model for Stokes–Brinkman flows with divergence-compatible velocity–pressure elements

Abstract

A two-level version for a recent semi-hybrid-mixed finite element approach for modeling Stokes and Brinkman flows is proposed. In the context of a domain decomposition of the flow region $\Omega$, composite divergence-compatible finite elements pairs in $H(\mathrm{div},\Omega )\times L^2\left(\Omega \right)$ are utilized for discretizing velocity and pressure fields, using the same approach previously adopted for two-level mixed Darcy and stress mixed elasticity models. The two-level finite element pairs of spaces in the subregions may have richer internal resolution than the boundary normal trace. Hybridization occurs by the introduction of an unknown (traction) defined over element boundaries, playing the role of a Lagrange multiplier to weakly enforce tangential velocity continuity and Dirichlet boundary condition. The well-posedness of the method requires a proper choice of the finite element space for the traction multiplier, which can be achieved after a proper velocity FE space enrichment with higher order bubble fields. The method is strongly locally conservative, yielding exact divergence-free velocity fields, demonstrating pressure robustness, and facilitating parallel implementations by limiting the communication of local common data to at most two elements. Easier coupling strategies of finite elements regarding different polynomial degree or mesh widths are permitted, provided that mild mesh and normal trace consistency properties are satisfied. Significant improvement in computational performance is achieved by the application of static condensation, where the global system is solved for coarse primary variables. The coarse primary variables are a piecewise constant pressure variable over the subregions, velocity normal trace and tangential traction over subdomain interfaces, as well as a real number used as a multiplier ensuring global zero-mean pressure. Refined details of the solutions are represented by secondary variables, which are post-processed by local solvers. Numerical results are presented for the verification of convergence histories of the method.