Mixed Methods for a Stream-Function – Vorticity Formulation of the Axisymmetric Brinkman Equations

Abstract

This paper is devoted to the numerical analysis of a family of finite element approximations for the axisymmetric, meridian Brinkman equations written in terms of the stream-function and vorticity. A mixed formulation is introduced involving appropriate weighted Sobolev spaces, where well-posedness is derived by means of the Babuska---Brezzi theory. We introduce a suitable Galerkin discretization based on continuous piecewise polynomials of degree $$k\ge 1$$kź1 for all the unknowns, where its solvability is established using the same framework as the continuous problem. Optimal a priori error estimates are derived, which are robust with respect to the fluid viscosity, and valid also in the pure Darcy limit. A few numerical examples are presented to illustrate the convergence and performance of the proposed schemes.