A semi‐hybrid‐mixed method for Stokes–Brinkman–Darcy flows with H(div)‐velocity fields

Abstract

SummaryWe consider new semi‐hybrid‐mixed finite element formulations for Stokes–Brinkman problems. Using ‐conforming approximate velocity fields, the continuity of normal components over element interfaces is taken for granted, and pressure is searched in discontinuous spaces preserving the divergence compatibility property. Tangential continuity is weakly imposed by a Lagrange multiplier playing the role of tangential traction. The method is strongly mass‐conservative, leading to exact divergence‐free simulations of incompressible flows. The Lagrange multiplier space requires specific choices according to the velocity approximations implemented in each element geometry. In certain cases, classic divergence‐compatible pairs adopted for Darcy flows may require divergence‐free bubble enrichment to enforce tangential continuity in some extent, avoiding any extra stabilization technique. Considerable improvement in computational performance is achieved by the application of static condensation: the global system is solved only for a piecewise constant pressure variable, velocity normal trace and tangential traction over interfaces. The remaining solution components are recovered by solving independent local Neumann problems in each element. Numerical results are presented for a set of standard test cases in the field of Stokes–Brinkman–Darcy flows with known analytical solutions. The main convergence properties of the method are verified in the whole range of parameters, from Stokes to Darcy limits, as well as for the combined Stokes–Darcy scenario. The robustness of the method is also demonstrated for more challenging test problems with complex non‐conforming meshes with local refinement patterns as well as for curved geometry.