GIMNAP, Departamento de Matemática, Facultad de Ciencias,
Universidad del Bío-Bío, Casilla
2 GIMNAP, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI 2MA), Universidad de Concepción, Concepción, Chile
3 Departamento de Matemáticas y Estadísticas, Universidad de Córdoba, Colombia
4 Institute of Earth Sciences, UNIL-Mouline Géopolis, University of Lausanne, 1015 Lausanne, Switzerland
Received: 22 July 2014
Revised: 29 January 2015
This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart−Thomas elements of arbitrary order k ≥ 0, piecewise continuous polynomials of degree k + 1, and piecewise polynomials of degree k, respectively. The well-posedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babuška–Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions.
Mathematics Subject Classification: 65N3065N1276D0765N1565J20
Key words: Brinkman equations / axisymmetric domains / augmented mixed finite elements / well-posedness analysis / error estimates
This work has been supported by CONICYT-Chile through: FONDECYT postdoctorado project No. 3120197, project Inserción de Capital Humano Avanzado en la Academia No. 79112012, FONDECYT regular project No. 1140791, and project Anillo ANANUM-ACT1118; by DIUBB through project 120808 GI/EF, and by the Swiss National Science Foundation through the research grant PP00P2144922.
© EDP Sciences, SMAI, 2015