Abstract
We present in this work an a posteriori error estimator for a porous media flow problem that follows the Brinkman model. First, we introduce the pseudostress as an auxiliary unknown, which let us to eliminate the pressure and thus derive a dual-mixed formulation in velocity-pseudostress. Next, in order to circumvent an inf-sup condition for the unique solvability, we stabilize the scheme by adding some appropriate least squares terms. The existence and uniqueness of solution are guaranteed and we derive an a posteriori error estimator based on the Ritz projection of the error, which is reliable and efficient up to high order terms. Finally, we report one numerical example confirming the good properties of the estimator.
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References
D.N. Arnold, R.S. Falk, Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials. Arch. Ration. Mech. Anal. 98(2), 143–165 (1987)
T.P. Barrios, R. Bustinza, An augmented discontinuous Galerkin method for stationary Stokes problem, Preprint 2010–20, Departamento de Ingeniería Matemática, Universidad de Concepción, 2010
T.P. Barrios, R. Bustinza, G.C. García, E. Hernández, On stabilized mixed methods for generalized Stokes problem based on the velocity-pseudostress formulation: a priori error estimates. Comput. Methods Appl. Mech. Eng. 237–240, 78–87 (2012)
T.P. Barrios, R. Bustinza, G.C. García, M. González, A posteriori error analyses of a velocity-pseudostress formulation of the generalized Stokes problem, Preprint 2013–04, Centro de Investigación en Ingeniería Matemática, Universidad de Concepción, 2013
Z. Cai, B. Lee, P. Wang, Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42, 843–859 (2004)
Z. Cai, C. Tong, P.S. Vassilevski, C. Wang, Mixed finite element methods for incompressible flow: stationary Stokes equations. Numer. Methods Partial Differ. Equ. 26, 957–978 (2010)
G.N. Gatica, A. Márquez, M.A. Sánchez, Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Eng. 199, 1064–1079 (2010)
G.N. Gatica, A. Márquez, M.A. Sánchez, A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows. Comput. Methods Appl. Mech. Eng. 200, 1619–1636 (2011)
G.N. Gatica, A. Márquez, M.A. Sánchez, Pseudostress-based mixed finite element methods for the Stokes problem in \(\mathbb{R}^{n}\) with Dirichlet boundary conditions. I: a priori error analysis. Commun. Comput. Phys. 12, 109–134 (2012)
G.N. Gatica, L.F. Gatica, A. Márquez, Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow. Numer. Math. 126(4), 635–677 (2014)
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Barrios, T., Bustinza, R., García, G.C., González, M. (2015). An a Posteriori Error Estimator for a New Stabilized Formulation of the Brinkman Problem. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_25
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DOI: https://doi.org/10.1007/978-3-319-10705-9_25
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