Hostname: page-component-89b8bd64d-72crv Total loading time: 0 Render date: 2026-05-07T14:54:21.798Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations

Published online by Cambridge University Press:  03 June 2015

Yalchin Efendiev ,
Juan Galvis ,
Guanglian Li  and
Michael Presho
Yalchin Efendiev*
Affiliation:
Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843, USA
Juan Galvis*
Affiliation:
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá D.C., Colombia
Guanglian Li*
Affiliation:
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843, USA
Michael Presho*
Affiliation:
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843, USA
*
Email:efendiev@math.tamu.edu
Email:jcgalvisa@unal.edu.co
Email:lotusli@math.tamu.edu
Corresponding author.Email:mpresho@math.tamu.edu
Get access

Abstract

In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.

Keywords

35J6065N30Generalized multiscale finite element methodnonlinear equationshigh-contrast

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 3 , March 2014 , pp. 733 - 755
Copyright
Copyright © Global Science Press Limited 2014

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.)

Article purchase

Temporarily unavailable

References

[1]Aarnes, J. E., On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation, SIAM Multiscale, J.Model. Simul., 2 (2004), pp. 421–439.Google Scholar
[2]Aarnes, J. E., Krogstad, S. and Lie, K.-A., A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform grids, Multiscale Model. Simul. 5(2) (2006), pp. 337–363.Google Scholar
[3]Abatangelo, L. and Terracini, S., Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent, Fixed, J.Point Theory Appl., 10 (2011), pp. 147–180.Google Scholar
[4]Adbulle, A. and Bai, Y., Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems, J. Comput. Phys., 231 (2012), pp. 7014–7036.Google Scholar
[5]Abdulle, A. and Vilmart, G., Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, to appear in Math. Comput., 2013.Google Scholar
[6]Albrecht, F., Haasdonk, B., Ohlberger, M. and Kaulmann, S., The localized reduced basis mul-tiscale method, Proceedings of Algorithmy, 2012.Google Scholar
[7]Arbogast, T., Pencheva, G., Wheeler, M. F. and Yotov, I., A multiscale mortar mixed finite element method, SIAM Multiscale, J.Model. Simul., 6(1) (2007), pp. 319–346.Google Scholar
[8]Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2001), pp. 1749–1779 (electronic).Google Scholar
[9]Babuska, I. and Lipton, R., Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, SIAM Multiscale Model. Simul., 9 (2011), pp. 373–406.Google Scholar
[10]Babus, I.ˇka and Melenk, J. M., The partition of unity method, Int. J. Numer. Method Eng., 40 (1997), pp. 727–758.Google Scholar
[11]Babuška, I. and Osborn, E., Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal., 20 (1983), pp. 510–536.Google Scholar
[12]Barrault, M., Maday, Y.,Nguyen, N. C. and Patera, A. T., An “Empirical Interpolation” method: application to efficient reduced-basis discretization of partial differential equations, CR Acad Sci Paris Series I, 339 (2004), pp. 667–672.Google Scholar
[13]Berlyand, L. and Owhadi, H., Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Arch. Ration. Mech. Anal., 198 (2010), pp. 677–721.CrossRefGoogle Scholar
[14]Boyaval, S., Reduced-basis approach for homogenization beyond periodic setting, SIAM MMS, 7(1) (2008), pp. 466–494.Google Scholar
[15]Boyaval, S., Le, C.Bris, Lelie, T.vre, Maday, Y., Nguyen, N. and Patera, A., Reduced basis techniques for stochastic problems, Arch. Comput. Method Eng., 17 (2010), pp. 435–454.Google Scholar
[16]Chaturantabut, S. and Sorensen, D., Nonlinear model reduction via discrete empirical interpolation, J. Sci. Comput., 32(5) (2010), pp. 2737–2764.Google Scholar
[17]Chen, Y. and Durlofsky, L., An ensemble level upscaling approach for efficient estimation of fine-scale production statistics using coarse-scale simulations, SPE paper 106086, presented at the SPE Reservoir Simulation Symposium, Houston, Feb. 26–28 (2007).Google Scholar
[18]Chen, Y., Durlofsky, L. J., Gerritsen, M., and Wen, X. H., A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations, Adv. Water Res., 26 (2003), pp. 1041–1060.Google Scholar
[19]Douglas Jr, J., Pacs-Leme, P. J. and Giorgi, T., Generalized forchheimer flow in porous media, in Boundary Value Problems for Partial Diff. Equations Appl., RMA Res. Notes, Appl. Math.,29, Masson, Paris, pp. 99–103, 1993.Google Scholar
[20]Drohmann, M., Haasdonk, B. and Ohlberger, M., Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation, J. Sci. Comput., 34(2) (2012), pp. A937–A969.Google Scholar
[21]Dryja, M., On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients, Comput. Methods Appl. Math., 3 (2003), pp. 76–85.Google Scholar
[22]Efendiev, Y., Galvis, J., Ki Kang, S., and Lazarov, R. D., Robust multiscale iterative solvers for nonlinear flows in highly heterogeneous media, Numer. Math. Theory Methods Appl., 5(3) (2012), 359173.CrossRefGoogle Scholar
[23]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, Huang, Y., Kornhuber, R., Widlund, O. and Xu, J. (Eds.), Volume 78 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Part 3, pp. 407–414, 2011.Google Scholar
[24]Efendiev, Y., Galvis, J., Lazarov, R., Moon, M. and Sarkis, M., Generalized Multiscale Finite Element Method, Symmetric Interior Penalty Coupling, In preparation.Google Scholar
[25]Efendiev, Y., Galvis, J., Lazarov, R., and Willems, J., Robust domain decomposition precondi-tioners for abstract symmetric positive definite bilinear forms, submitted.Google Scholar
[26]Efendiev, Y., Galvis, J., and Hou, T., Generalized multiscale finite element methods, submitted at http://arxiv.org/submit/631572.Google Scholar
[27]Efendiev, Y., Galvis, J., and Thomines, F., A systematic coarse-scale model reduction technique for parameter-dependent flows in highly heterogeneous media and its applications, 2011Google Scholar
[28]Efendiev, Y., Galvis, J., and Wu, X. H., Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comput. Phys., 230(4) (2011), pp. 937–955.Google Scholar
[29]Efendiev, Y., Ginting, V., Hou, T. and Ewing, R., Accurate multiscale finite element methods for two-phase flow simulations, J. Comput. Phys., 220(1) (2006), pp. 155–174.Google Scholar
[30]Efendiev, Y. and Hou, T., Multiscale Finite Element Methods, Theory and Applications, Springer, 2009.Google Scholar
[31]Efendiev, Y., Hou, T., and Ginting, V., Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci., 2 (2004), pp. 553–589.Google Scholar
[32]Elfverson, D., Georgoulis, E., Ma, A.˚lqvist and Peterseim, D., Convergence of a discontinuous galerkin multiscale method, arxiv: 1211.5524, 2012.Google Scholar
[33]Eyck, A. and Lew, A., Discontinuous Galerkin methods for nonlinear elasticity, Numer, Int. J.Meth. Eng., 67(9) (2006), pp. 1204–1243.Google Scholar
[34]Galvis, J. and Efendiev, Y., Domain decomposition preconditioners for multiscale flows in high-contrast media: reduced dimension coarse spaces, SIAM Multiscale, J.Model. Simul., 8(5) (2011), pp. 1621–1644.Google Scholar
[35]Givler, R. C. and Altobelli, S. A., A determination of the effective viscosity for the Brinkman-Forchheimer flow model, Fluid, J.Mech., 258 (1994), pp. 355–370.Google Scholar
[36]Grepl, M. A., Maday, Y., Nguyen, N. C. and Patera, A. T., Efficient reduced-basis treatment of non-affine and nonlinear partial differential equations, ESIAM: M2AN, 41(2) (2007), pp. 575–605.Google Scholar
[37]Henning, P., Målqvist, A., and Peterseim, D., A rigorous multiscale method for semi-linear elliptic problems, arxiv: 1211.3551, 2012.Google Scholar
[38]Hou, T. Y. and Wu, X. H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169–189.CrossRefGoogle Scholar
[39]Hughes, T., Feijoo, G., Mazzei, L., and Quincy, J., The variational multiscale method–a paradigm for computational mechanics, Comput. Methods Appl. Mech. Eng., 166 (1998), pp. 3–24.Google Scholar
[40]Huynh, D., Knezevic, D. and Patera, A., A static condensation reduced basis element method: approximation and a posteriori error estimation, ESAIM: M2AN, 47 (2013), pp. 213–251.Google Scholar
[41]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.Google Scholar
[42]Machiels, L., Maday, Y., Oliveira, I. B., Patera, A. T., and Rovas, D. V., Output bounds for reduced-basisapproximationsof symmetric positive definite eigenvalue problems, Comptes Rendus de l’Académie des Sciences, 331(2) (2000), pp. 153–158.Google Scholar
[43]Maday, Y., Reduced-basis method for the rapid and reliable solution of partial differential equations, Proceedings of International Conference of Mathematicians, Madrid. European Mathematical Society, Zurich, 2006.Google Scholar
[44]Maday, Y. and Rønquist, E., A reduced-basis element method, J. Sci. Comput., 17(1-4) (2002), pp. 447–459.Google Scholar
[45]Nguyen, N. C., A multiscale reduced-basis method for parameterized elliptic partial differential equations with multiple scales, J. Comput. Phys., 227(23) (2008), pp. 9807–9822.Google Scholar
[46]Ohlberger, M. and Schaefer, M., Error control based model reduction for parameter optimization of elliptic homogenization problems, Proceedings of the 1st IFAC Workshop on Control of Systems Governed by PDE’s, 2013.Google Scholar
[47]Richards, L., Capillary conduction of liquids through porous mediums, Physics, pp. 318–333, 1931.Google Scholar
[48]Rivière, B., Discontinuous Galerkin methods for solving elliptic and parabolic equation, Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), 35, Philadelphia, 2008.Google Scholar
[49]Rozza, G., Huynh, D. B. P, and Patera, A. T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Application to transport and continuum mechanics, Arch. Comput. Method. Eng., 15(3) (2008), pp. 229–275.Google Scholar
[50]Ruth, D. and Ma, H., On the derivation of the Forchheimer equation by means of the averaging theorem, Trans. Por. Med., 7 (1992), pp. 255–264.Google Scholar