Nonconforming polynomial mixed finite element for the Brinkman problem over quadrilateral meshes

Abstract

This work provides a new mixed finite element method for the Brinkman problem over arbitrary convex quadrilateral meshes. The velocity is approximated by piecewise polynomial element space which is H(div)-nonconforming, and the pressure is approximated by piecewise constant. We give the convergence analysis of our element, and especially show the robustness with respect to the Darcy limit. Moreover, via a discrete de Rham complex, a higher-order approximation error term is obtained for incompressible flow. Numerical examples verify our theoretical findings.