Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T00:03:37.042Z Has data issue: false hasContentIssue false
  • English
  • Français

Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

Published online by Cambridge University Press:  28 May 2015

Y. Efendiev ,
J. Galvis ,
S. Ki Kang  and
R.D. Lazarov
Y. Efendiev*
Affiliation:
Department of Mathematics, TAMU, College Station, Texas, 77843, USA
J. Galvis*
Affiliation:
Department of Mathematics, TAMU, College Station, Texas, 77843, USA
S. Ki Kang*
Affiliation:
Department of Mathematics, TAMU, College Station, Texas, 77843, USA
R.D. Lazarov*
Affiliation:
Department of Mathematics, TAMU, College Station, Texas, 77843, USA
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards’ equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.

Keywords

65M6035Q3535Q8665N3065N55FE methodnonlinear permeabilityhighly heterogeneous mediahigh contrast media
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 3 , August 2012 , pp. 359 - 383
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abdulle, A. and Vilmart, G., A priori error estimates f or finite element methods with numerical quadrature for nonmonotone non linear elliptic problems, Submitted for publication.Google Scholar
[2]Aarnes, J. and Hou, T., Multiscale domain decomposition methods for elliptic problems with high aspect ratios, Acta Math. Appl. Sin. Engl. Ser., 18(1), 2002, pp. 63–76.CrossRefGoogle Scholar
[3]Arbogast, T., Wheeler, M.F., and Zhang, N.Y., A nonlinear mixed finite element method for a degenerate parabolic equationarising in flow in porous media, SIAM J. Numer. Anal. 33 (4), 1996, pp. 1669–1687.CrossRefGoogle Scholar
[4]Bergh, J. and Löfstrom, J., Interpolation Spaces, an Introduction, Springer-Verlag, Berlin, 1976.CrossRefGoogle Scholar
[5]Brenner, S.C. and L.Scott, R., The Mathemethical Theory of Finite Element Methods, Springer-Verlag, 1994.CrossRefGoogle Scholar
[6]Cai, X.C. and D.Keyes, E., Nonlinearly preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 24, 2002, pp. 183–200.CrossRefGoogle Scholar
[7]Cai, X.C., Marcinkowski, L., and Vassilevski, P., An element agglomeration nonlinear additive Schwarz preconditioned Newton method for unstructured finite element problems, Appl. Math., 50, no. 3, 2005, pp. 247–275.CrossRefGoogle Scholar
[8]Chavent, G. and Jaffré, J., Mathematical Models and Finite Elements for Reservoir Simulation; Single Phase, Multiphase and Multicomponent Flows through Porous Media, Elsevier Science Publishers, North-Holland, Amsterdam, New York, Oxford, Tokyo, 1986.Google Scholar
[9]Chen, Z., Deng, W.B., and Ye, H., Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media, Communications in Mathematical Sciences 3 (2005), 493–515.CrossRefGoogle Scholar
[10]Ciarlet, P.G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, 1988.Google Scholar
[11]Dryja, M. and Hackbusch, W, On the nonlinear domain decomposition method, BIT Numerical Mathematics, Volume 37, Number 2, pp. 296–311, DOI: 10.1007/BF02510214CrossRefGoogle Scholar
[12]Dugundgi, J., Topology, Allyn and Bacon, Boston, 1966Google Scholar
[13]Efendiev, Y. and Hou, T., Multiscale Finite Element Methods. Theory and Applications, Springer, 2009.Google Scholar
[14]Efendiev, Y., Hou, T. and V.|Ginting, Multiscale finite element methods for nonlinear partial differential equations, Comm. Math. Sci., 2(4), 2004Google Scholar
[15]Efendiev, Y., Hou, TY, and Wu, X.H., Convergence of a nonconforming multiscale finite element method, SIAM J. Num. Anal., 37 2000, pp. 888–910.CrossRefGoogle Scholar
[16]Efendiev, Y. and Galvis, J., Domain decomposition preconditioner for multiscale high-contrast problems, in: Huang, Y, Kornhuber, R., Widlund, O., Xu, J. (Eds.), Domain Decomposition Methods in Science and Engineering XIX, Volume 78 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2011, pp. 189–196.CrossRefGoogle Scholar
[17]Efendiev, Y., Galvis, J., and Vassielvski, P., Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic Problems with High-Contrast Coefficients, in Domain Decomposition Methods in Science and Engineering XIX, Y, Huang; R., Kornhuber; O., Widlund; J., Xu (Eds.), Volume 78 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, 2011, Part 3, pp. 407–414.Google Scholar
[18]Galvis, J., and Efendiev, Y., Domain decomposition preconditioners for multiscale flows in high contrast media, Multiscale Model. Simul. 8, 2010, pp. 1461–1483.CrossRefGoogle Scholar
[19]Galvis, J., and Efendiev, Y., Domain decomposition preconditioners for multiscale flows in high contrast media. Reduced dimension coarse spaces, Multiscale Model. Simul. 8, 2010, pp. 1621–1644.CrossRefGoogle Scholar
[20]Graham, I.G., Lechner, P.O., and Scheichl, R., Domain decomposition for multiscale PDEs, Numer. Math., 106(4), 2007, pp. 589–626.CrossRefGoogle Scholar
[21]Graham, I.G. and Scheichl, R., Robust domain decomposition algorithms for multiscale PDEs, Numer. Methods Partial Differential Equations, 23(4), 2007, pp. 859–878.CrossRefGoogle Scholar
[22]Haverkamp, R., Vauclin, M., Touma, J., Weirenga, P., and Vachaud, G., Comparison of numerical simulation models for one-dimensional infiltration, Soil Sci. Soc. Am. J., 41, 1977, pp. 285–294.CrossRefGoogle Scholar
[23]Hou, T.Y. and X.Wu, H., A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics, 134, 1997, pp. 169–189.CrossRefGoogle Scholar
[24]Jenny, P., Lee, S.H., and Tchelepi, H., Multi-scale finite volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187, 2003, pp. 47–67.CrossRefGoogle Scholar
[25]Kim, T., Pasciak, J., and Vassilevski, P., Mesh-independent convergence of the modified inexact Newton method for a second order non-linear problem, Numer. Linear Algebra Appl., 13,no. 1, 2006, pp. 23–47.CrossRefGoogle Scholar
[26]Mathew, T.P.A., Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, volume 61 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2008.CrossRefGoogle Scholar
[27]Meyers, N.G., An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Annali dela Scuola Normale Superiore di Pisa, Ser. III, XVII, 1963, pp. 189–206.Google Scholar
[28]Panfilov, M., Macroscale models of flow through highly heterogeneous porous media, Springer Netherlands, 2000.CrossRefGoogle Scholar
[29]Radu, F.A., Pop, I.S., and Knabner, P., Error estimates for a mixed finite element discretization of some degenerate parabolic equations, Numer. Math. 109 (2), 2008, pp. 285–311.CrossRefGoogle Scholar
[30]Rajagopal, S.K., On a hierarchy of approximate models for flows in incompressible fluids through porous solids, Math. Models Methods Appl. Sci., 17, 2007, pp. 215–252.CrossRefGoogle Scholar
[31]Richards, L., Capillary conduction of liquids through porous mediums, Physics, 1931, pp. 318–333.CrossRefGoogle Scholar
[32]Salgado, A.J., Approximation techniques for incompressible flows with heterogeneous properties, PhD. Thesis, Texas A&M University, 2010.Google Scholar
[33]Simader, C.G., On Dirichlet’s Boundary Value Problem, Lecture Notes in three dimensions. Nu-mer.Math. 60, 1991, pp. 219–234.Google Scholar
[34]Tai, X.C. and Espedal, M., Applications of a space decomposition method to linear and nonlinear elliptic problems, Numer. Methods Partial Differential Equations 14, no. 6, 1998, pp. 717–737.3.0.CO;2-T>CrossRefGoogle Scholar
[35]Toselli, A. and Widlund, O., Domain decomposition methods—algorithms and theory, volume 34 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2005.CrossRefGoogle Scholar
[36]Vangenuchten, M.Th., A closed form equations for predicting the hydraulic conductivity of un-saturated soils, Soil Sci. Soc. Am. J., 44, 1980, pp. 892–898.CrossRefGoogle Scholar
[37]Xu, J., Two-grid discretization techniques for linear and nonlinear PDEs, SIAM Journal on Numerical Analysis 33, 1996, pp. 1759–1777CrossRefGoogle Scholar
[38]Xu, J. and Zikatanov, L., On an energy minimizing basis for algebraic multigrid methods. Comput. Visual. Sci., 7, 2004, pp. 121–127.CrossRefGoogle Scholar