Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

In this paper we develop and analyze a unified approximation of the velocity–pressure pair for the Stokes–Darcy coupled problem in a plane domain. It is well known that, stable finite element approximations for the Stokes problem may not be appropriate for Darcy problem and for the coupling of fluid flow (modeled by the Stokes equations) with porous media flow (modeled by the Darcy equation), and therefore, different spaces are commonly used for the discretizations of the Darcy and the Stokes problems. In this work we proposed a modification of the Darcy problem which allows us to apply the classical Mini-element to the whole coupled Stokes–Darcy problem. The proposed method is probably one of the cheapest method for continuous approximation of the coupled system, has optimal accuracy with respect to solution regularity, and has simple and straightforward implementations. Numerical experiments are also presented, which confirm the excellent stability and accuracy of our method. © 2018 Elsevier Ltd

Registro:

Documento: Artículo
Título:A unified mixed finite element approximations of the Stokes–Darcy coupled problem
Autor:Armentano, M.G.; Stockdale, M.L.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina; IMAS - Conicet, Buenos Aires, 1428, Argentina
Palabras clave:Mixed finite elements; Stability analysis; Stokes–Darcy problem; Finite element method; Navier Stokes equations; Numerical methods; Porous materials; Continuous approximations; Finite element approximations; Mixed finite element approximation; Mixed finite elements; Numerical experiments; Optimal accuracy; Porous-media flow; Stability analysis; Flow of fluids
Año:2019
Volumen:77
Número:9
Página de inicio:2568
Página de fin:2584
DOI: http://dx.doi.org/10.1016/j.camwa.2018.12.032
Título revista:Computers and Mathematics with Applications
Título revista abreviado:Comput Math Appl
ISSN:08981221
CODEN:CMAPD
Registro:http://digital.bl.fcen.uba.ar/collection/paper/document/paper_08981221_v77_n9_p2568_Armentano

Referencias:

  • Araya, R., Barrenechea, G.R., Poza, A., An adaptive stabilized finite element method for the generalized Stokes problem (2008) J. Comput. Appl. Math., 214, pp. 457-479
  • Armentano, M.G., Blasco, J., Stable and unstable cross-grid P k Q l mixed finite elements for the Stokes problem (2010) J. Comput. Appl. Math., 234 (5), pp. 1404-1416
  • Badia, S., Codina, R., Stokes, Maxwell and Darcy: A single finite element approximation for three model problems (2012) Appl. Numer. Math., 62, pp. 246-263
  • Bochev, P.B., Dohrmann, C.R., Gunzburger, M.D., Stabilization of low-order mixed finite elements for the Stokes equations (2006) SIAM J. Numer. Anal., 44 (1), pp. 82-101
  • Boffi, D., Minimal stabilizations of the P k +1-P k approximation of the stationary Stokes equations (1995) Math. Models Methods Appl. Sci., 5 (2), pp. 213-224
  • Boffi, D., Brezzi, F., Demkowicz, L., Durán, R.G., Falk, R., Fortin, M., Mixed finite elements, compatibility conditions, and applications (2008) Lect. Notes Math., 1939
  • Boffi, D., Gastaldi, L., On the quadrilateral Q 2 -P 1 element for the Stokes problem (2002) Internat. J. Numer. Methods Fluids, 39 (4), pp. 1001-1011
  • Brezzi, F., Falk, R., Stability of higher-order Hood-Taylor methods (1991) SIAM J. Numer. Anal., 28 (3), pp. 581-590
  • Brezzi, F., Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer Berlin Heidelberg New York; Barrenechea, G.R., Franca, L.P., Valentin, F., A Petrov–Galerkin enriched method: A mass conservative finite element method for the Darcy equation (2007) Comput. Methods Appl. Mech. Engrg., 196, pp. 2449-2464
  • Bochev, P.B., Dohrmann, C.R., A computational study of stabilized C0, low-order finite element approximations of Darcy equations (2006) Comput. Mech., 38, pp. 323-333
  • Gatica, G.N., Ruiz-Baier, R., Tierra, G., Giordano A mixed finite element method for Darcy's equations with pressure dependent porosity (2016) Math. Comp., 85 (297), pp. 1-33
  • Gatica, G.N., Meddahi, S., Oyarzúa, R., A conforming mixed finite-element method for the coupling of fluid flow with porous media flow (2009) IMA J. Numer. Anal., 29, pp. 86-108
  • Gatica, G.N., Oyarzúa, R., Sayas, F.J., Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem (2011) Math. Comp., 80 (276), pp. 1911-1948
  • Layton, W.J., Schieweck, F., Yotov, I., Coupling fluid flow with porous media flow (2003) Siam J. Numer. Anal., 40 (6), pp. 2195-2218
  • Nicaise, N.S., Ahounou, B., Houedanou, W., Residual-based a posteriori error estimates for a nonconforming finite element discretization of the Stokes-Darcy coupled problem: isotropic discretization (2016) Afr. Mat., 27 (3-4), pp. 701-729
  • Rui, H., Zhang, R., A unified stabilized mixed finite element method for coupling Stokes and Darcy flows (2009) Comput. Methods Appl. Mech. Engrg., 198, pp. 2692-2699
  • Discacciati ans A. Quarteroni, M., Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation (2009) Rev. Math. Comput., 22, pp. 315-426
  • Karper, T., Mardal, K.A., Winther, R., Unified Finite Element Discretizations Of Coupled Darcy-Stokes Flow (2009) Numer. Methods Partial Differential Equations, 25 (2), pp. 311-326
  • Brezzi, F., Fortin, M., Marini, L.D., Mixed Finite Element Methods with continuous stresses (1993) Math. Models Methods Appl. Sci., 3 (2), pp. 275-287
  • Beavers, G., Joseph, D., Boundary conditions at a naturally impermeable wall (1967) J. Fluid Mech., 30, pp. 197-207
  • Girault, V., Raviart, P.A., (1986) Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5. , Springer-Verlag
  • Raviart, P.A., Thomas, J.M., (1977) A Mixed Finite Element Method for Second Order Elliptic Problems, Mathematical Aspects Of the Finite Element Method, Lectures Notes in Math., 606. , Galligani I. Magenes E. Springer Verlag
  • Arnold, D.N., Brezzi, F., Fortin, M., A stable finite element for the Stokes equations (1984) Calcolo, 21, pp. 337-344
  • Braess, D., Finite elements: Theory, fast solvers, and applications in solid mechanics (2007), Cambridge University Press; Ciarlet, P.G., Lions, J.L., Handbook of Numerical Analysis, Vol. 8 (1990), Gulf Professional Publishing; Weiber, S., Residual error estimate for BEM-based FEM on polygonal meshes (2011) Numer. Math., 118 (4), pp. 765-788
  • Verfürth, R., A Posteriori Error Estimation Techniques for Finite Element Methods, Numerical Mathematics and Scientific Computation (2013), Oxford University Press Oxford

Citas:

---------- APA ----------
Armentano, M.G. & Stockdale, M.L. (2019) . A unified mixed finite element approximations of the Stokes–Darcy coupled problem. Computers and Mathematics with Applications, 77(9), 2568-2584.
http://dx.doi.org/10.1016/j.camwa.2018.12.032
---------- CHICAGO ----------
Armentano, M.G., Stockdale, M.L. "A unified mixed finite element approximations of the Stokes–Darcy coupled problem" . Computers and Mathematics with Applications 77, no. 9 (2019) : 2568-2584.
http://dx.doi.org/10.1016/j.camwa.2018.12.032
---------- MLA ----------
Armentano, M.G., Stockdale, M.L. "A unified mixed finite element approximations of the Stokes–Darcy coupled problem" . Computers and Mathematics with Applications, vol. 77, no. 9, 2019, pp. 2568-2584.
http://dx.doi.org/10.1016/j.camwa.2018.12.032
---------- VANCOUVER ----------
Armentano, M.G., Stockdale, M.L. A unified mixed finite element approximations of the Stokes–Darcy coupled problem. Comput Math Appl. 2019;77(9):2568-2584.
http://dx.doi.org/10.1016/j.camwa.2018.12.032