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Pseudostress-Based Mixed Finite Element Methods for the Stokes Problem in ℝn with Dirichlet Boundary Conditions. I: A Priori Error Analysis

Published online by Cambridge University Press:  20 August 2015

Gabriel N. Gatica ,
Antonio Márquez  and
Manuel A. Sánchez
Gabriel N. Gatica*
Affiliation:
CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Antonio Márquez*
Affiliation:
CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Manuel A. Sánchez*
Affiliation:
CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
*
Corresponding author.Email:ggatica@ing-mat.udec.cl
Email:amarquez@uniovi.es
Email:manusanchez@udec.ci
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Abstract

We consider a non-standard mixed method for the Stokes problem in ℝn, n Є {2,3}, with Dirichlet boundary conditions, in which, after using the incompressibility condition to eliminate the pressure, the pseudostress tensor σ and the velocity vector u become the only unknowns. Then, we apply the Babuška-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k≥0 for σ and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme. In addition, we introduce and analyze an augmented approach for our pseudostress-velocity formulation. The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. For instance, Raviart-Thomas elements of order k≥0 for σ and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case. Finally, extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature, are provided.

Keywords

65N1565N3065N5074B05Mixed finite elementpseudostressincompressible flow
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 109 - 134
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Arnold, D.N., Douglas, J. and Gupta, Ch.P., A family of higher order mixed finite element methods for plane elasticity. Numerische Mathematik, vol. 45, pp. 1–22, (1984).Google Scholar
[2]Bochev, P.B. and Gunzburger, M.D., Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations. Computer Methods in Applied Mechanics and Engineering, vol. 126, 3-4, pp. 267–287, (1995).Google Scholar
[3]Bochev, P.B. and Gunzburger, M.D., Finite element methods of least-squares type. SIAM Review, vol. 40, 4, pp. 789–837, (1998).Google Scholar
[4]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.Google Scholar
[5]Cai, Z., Lee, B. and Wang, P., Least-squares methods for incompressible Newtonian fluid flow: Linear stationary problems. SIAM Journal on Numerical Analysis, vol. 42, 2, pp. 843–859, (2004).Google Scholar
[6]Cai, Z., Manteuffel, T.A. and McCormick, S.F., First-order system least squares for the Stokes equations, with application to linear elasticity. SIAM Journal on Numerical Analysis, vol. 34, 5, pp. 1727–1741, (1997).Google Scholar
[7]Cai, Z. and Starke, G., Least-squares methods for linear elasticity. SIAM Journal on Numerical Analysis, vol. 42, 2, pp. 826–842, (2004).Google Scholar
[8]Cai, Z., Tong, Ch., Vassilevski, P.S. and Wang, Ch., Mixed finite element methods for in-compressible flow: stationary Stokes equations. Numerical Methods for Partial Differential Equations, vol. 26, 4, pp. 957–978, (2009).Google Scholar
[9]Cai, Z. and Wang, Y., A multigrid method for the pseudostress formulation of Stokes problems. SIAM Journal on Scientific Computing, vol. 29, 5, pp. 2078–2095, (2007).Google Scholar
[10]Ciarlet, P.G., The Finite Element Method for Elliptic Problems. North-Holland, 1978.Google Scholar
[11]Figueroa, L., Gatica, G.N. and Heuer, N., A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows. Computer Methods in Applied Mechanics and Engineering, vol. 198, 2, pp. 280–291, (2008).Google Scholar
[12]Figueroa, L., Gatica, G.N. and Márquez, A., Augmented mixed finite element methods for the stationary Stokes Equations. SIAM Journal on Scientific Computing, vol. 31, 2, pp. 1082–1119, (2008).Google Scholar
[13]de Veubeke, B.M. Fraejis, Stress function approach. In: Proceedings of the World Congress on Finite Element Methods in Structural Mechanics, vol. 1, Bournemouth, Dorset, England (October 12-17, 1975), J.1 - J.51.Google Scholar
[14]Gatica, G.N., An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions. Electronic Transactions on Numerical Analysis, vol. 26, pp. 421–438, (2007).Google Scholar
[15]Gatica, G.N., Analysis of a new augmented mixed finite element method for linear elasticity allowing RT0–P1–P0 approximations. ESAIM Mathematical Modelling and Numerical Analysis, vol. 40, 1, pp. 1–28, (2006).Google Scholar
[16]Gatica, G.N., González, M. and uMeddahi, S., A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. I: A priori error analysis. Computer Methods in Applied Mechanics and Engineering, vol. 193, 9-11, pp. 881–892, (2004).Google Scholar
[17]Gatica, G.N., Márquez, A. and Meddahi, S., An augmented mixed finite element method for 3D linear elasticity problems. Journal of Computational and Applied Mathematics, vol. 231, 2, pp. 526–540, (2009).Google Scholar
[18]Gatica, G.N., Márquez, A. and Sánchez, M.A., Analysis of a velocity-pressure-pseudostress formulation for incompressible flow. Computer Methods in Applied Mechanics and Engineering, vol. 199, 17-20, pp. 1064–1079, (2010).Google Scholar
[19]Gatica, G.N., Márquez, A. and Sánchez, M.A., A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows. Computer Methods in Applied Mechanics and Engineering, vol. 200, 17-20, pp. 1619–1636, (2011).Google Scholar
[20]Girault, V. and Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, 1986.Google Scholar
[21]Hiptmair, R., Finite elements in computational electromagnetism. Acta Numerica, vol. 11, pp. 237–339, (2002).Google Scholar
[22]Howell, J.S., Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients. Journal of Computational and Applied Mathematics, vol. 231, 2, pp. 780–792, (2009).Google Scholar
[23]Roberts, J.E. and Thomas, J.M., Mixed and Hybrid Methods. In: Handbook of Numerical Analysis, edited by Ciarlet, P.G. and Lions, J.L., vol. II, Finite Element Methods (Part 1), 1991, North-Holland, Amsterdam.Google Scholar
[24]Tsai, C.C.Young, D.L.Cheng, A.H.-D., Meshless BEM for three-dimensional Stokes flows. Computer Modeling in Engineering and Sciences, vol. 3, 1, pp. 117–128, (2002).Google Scholar
[25]Young, D.L., Liu, Y.H., Eldho, T.I., Three-dimensional Stokes flow solution using combined boundary element and finite element methods. Chinese Journal of Mechanics, vol. 15, pp. 169–176, (1999).Google Scholar
[26]Young, D.L., Jane, S.C., Lin, C.Y., Chiu, C.L., Chen, K.C., Solutions of 2D and 3D Stokes laws using multiquadrics method. Engineering Analysis with Boundary Elements, vol. 28, 10, pp. 1233–1243, (2004).Google Scholar