Geodesic
Stabilized mixed approximation of axisymmetric Brinkman flows
Anaya, Verónica1 ; Mora, David2 ; Reales, Carlos3 ; Ruiz-Baier, Ricardo4
1 GIMNAP, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
2 GIMNAP, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI2MA), 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
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 855-874

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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.

Reçu le :
DOI : 10.1051/m2an/2015011
Classification : 65N30 65N12 76D07 65N15 65J20
Keywords: Brinkman equations, axisymmetric domains, augmented mixed finite elements, well-posedness analysis, error estimates

Anaya, Verónica 1 ; Mora, David 2 ; Reales, Carlos 3 ; Ruiz-Baier, Ricardo 4

1 GIMNAP, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
2 GIMNAP, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI2MA), 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 
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     title = {Stabilized mixed approximation of axisymmetric {Brinkman} flows},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {855--874},
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Anaya, Verónica; Mora, David; Reales, Carlos; Ruiz-Baier, Ricardo. Stabilized mixed approximation of axisymmetric Brinkman flows. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 855-874. doi: 10.1051/m2an/2015011

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