Abstract
Mixed-hybrid finite element discretization of the Darcy’s law and the continuity equation that describe the potential fluid flow problem in porous media leads to symmetric but indefinite linear systems for the velocity and pressure vector components. In this contribution we compare the computational efficiency of two main techniques for the solution of such systems based on a cheap elimination of a portion of variables and on subsequent iterative solution of a transformed system. We consider Schur complement reduction and null-space based projection. Since for both approaches the asymptotic convergence factor in the iterative part depends linearly on the mesh size parameter, we perform computational experiments on several real-world problems and report a comparison of numerical results which includes not only the cost of iterative part but also the overhead of initial transformation and back substitution process.
This work was supported by the Grant Agency of the Czech Republic under grant No. 101/00/1035 and by the Grant Agency of the Academy of Sciences of the Czech Republic under grant No. A1030103.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
M. Arioli, J. Maryška, M. Rozložník, and M. Tůma. The dual variable approach for the mixed-hybrid approximation of the potential fluid flow problem, Technical Report RAL-TR-2001-023, 2001.
F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods, Springer, 1991.
A. Greenbaum. Iterative Methods for Solving Linear Systems, SIAM, 1997.
E. F. Kaasschieter and A. J. M. Huijben. Mixed-hybrid finite elements and stream line computation for the potential flow problem, Numerical Methods for Partial Differential Equations, 8, 221–266, 1992.
J. Maryška, M. Rozložník, and M. Tůma. Mixed-hybrid finite element approximation of the potential fluid flow problem, J. Comput. Appl. Math., 63, 383–392, 1995.
J. Maryška, M. RozložnÞk, and M. Tůma. The potential fluid flow problem and the convergence rate of the minimal residual method, Numer. Linear Algebra with Applications, 3(6), 525–542, 1996.
J. Maryška, M. Rozložník, and M. Tůma. Schur complement systems in the mixed hybrid finite element approximation of the potential fluid flow problem, SIAM J. Sc. Comput., 22(2), 704–723, 2000.
C. C. Paige and M. A. Saunders. Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12, 617–629, 1975.
I. Perugia, V. Simoncini, and M. Arioli. Linear algebra methods in a mixed approximation of magnetostatic problems, SIAM J. Sci. Comput., 21, 1085–1101, 1999.
T. Rusten and R. Winther. A preconditioned iterative method for saddle point problems, SIAM J. Math. Anal. Appl., 13, 887–905, 1992.
A. J. Wathen, B. Fischer, and D. J. Silvester. The convergence rate of the minimal residual method for the Stokes problem, Num. Mathematik, 71, 121–134, 1995.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Maryška, J., Rozložník, M., Tůum, M. (2001). Primal vs. Dual Variable Approach for Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem in Porous Media. In: Margenov, S., Waśniewski, J., Yalamov, P. (eds) Large-Scale Scientific Computing. LSSC 2001. Lecture Notes in Computer Science, vol 2179. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45346-6_44
Download citation
DOI: https://doi.org/10.1007/3-540-45346-6_44
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43043-8
Online ISBN: 978-3-540-45346-8
eBook Packages: Springer Book Archive