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

Journal article

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 Babuška–Brezzi theory. We introduce a suitable Galerkin discretization based on continuous piecewise polynomials of degree (Formula presented.) 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.

Authors: Anaya, V; Mora, D; Reales, C; Ruiz-Baier, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer Verlag
• Journal: Journal of Scientific Computing
• Volume: 71
• Issue: 1
• Pages: 348–364
• Publication date: 2016-10-05
• Acceptance date: 2016-09-28
• DOI: 10.1007/s10915-016-0302-x
• EISSN: 1573-7691
• ISSN: 0885-7474
• Keywords: Stream-function and vorticity formulation; Stability analysis; Brinkman equations; Finite element method; Error estimates; Axisymmetric domains