High accuracy analysis of a nonconforming rectangular finite element method for the Brinkman model

We propose a simple nonconforming rectangular finite element method for the Brinkman model. The velocity space is edge-oriented, in which the local space of each component is P_2 plus the span of a cubic monomial, and the pressure space is piecewise linear. We prove that, if the mesh is uniform, this method is uniformly convergent with respect to the given parameters, with the convergence order O(h^2) in a mesh- and parameter-dependent norm for the velocity and in L^2 -norm for the pressure. Moreover, for the Stokes problem we show the velocity convergence order in L^2 -norm reaches O(h^3) provided the solution and the domain are sufficiently regular. The key idea is based on the consistency error estimates by a tangent-normal switching strategy. Numerical tests confirm our theory.