Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data

Authors

  • Jerome Droniou School of Mathematical Sciences, Monash University.

DOI:

https://doi.org/10.21914/anziamj.v56i0.9365

Keywords:

elliptic equations, parabolic equations, non-linear equations, convergence analysis, discrete functional analysis, compactness, non-smooth data

Abstract

We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming non-physical regularity of the data. For simplicity of exposure, we mostly consider linear elliptic equations, and we briefly explain how these techniques are adapted and extended to non-linear time-dependent meaningful models (Navier--Stokes equations, flows in porous media, etc.). These convergence techniques rely on Sobolev norms and a discrete form of functional analysis results. References
  • B. Andreianov, F. Boyer, and F. Hubert. Discrete duality finite volume schemes for Leray–Lions-type elliptic problems on general 2D meshes. Numer. Meth. Part. D. E., 23(1):145–195, 2007. doi:10.1002/num.20170
  • J-.P. Aubin. Un theoreme de compacite. C. R. Math. Acad. Sci. Paris, 256:5042–5044, 1963. http://gallica.bnf.fr/ark:/12148/bpt6k4006n/f1164.item.r=.zoom
  • P. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, and J. L. Vazquez. An \(L^1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22(2):241–273, 1995. http://www.numdam.org/item?id=ASNSP_1995_4_22_2_241_0
  • H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. doi:10.1007/978-0-387-70914-7
  • F. Brezzi, K. Lipnikov, and V. Simoncini. A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Meth. Appl. Sci. 15(10):1533–1551, 2005. doi:10.1142/S0218202505000832
  • C. Chainais-Hillairet and J. Droniou. Convergence analysis of a mixed finite volume scheme for an elliptic-parabolic system modeling miscible fluid flows in porous media. SIAM J. Numer. Anal. 45(5):2228–2258, 2007. doi:10.1137/060657236
  • C. Chainais-Hillairet, S. Krell, and A. Mouton. Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media. Numer. Meth. Part. D. E., 31(3):723–760, 2015. doi:10.1002/num.21913
  • E. Chenier, R. Eymard, T. Gallouet, and R. Herbin. An extension of the MAC scheme to locally refined meshes: convergence analysis for the full tensor time-dependent Navier–Stokes equations. Calcolo 52(1):69–107, 2015. doi:10.1007/s10092-014-0108-x
  • M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. Rev. Francaise Automat. Informat. Recherche Operationnelle Ser. Rouge, 7(R-3):33–75, 1973. https://eudml.org/doc/193250
  • G. Dal Maso, F. Murat, L. Orsina, and A. Prignet. Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28(4):741–808, 1999. http://www.numdam.org/item?id=ASNSP_1999_4_28_4_741_0
  • D. A. Di Pietro and M. Vohralik. A review of recent advances in discretization methods, a posteriori error analysis, and adaptive algorithms for numerical modeling in geosciences. Oil Gas Sci. Technol. 69(4):701–730, 2014. doi:10.2516/ogst/2013158
  • D. A. Di Pietro and A. Ern. Mathematical aspects of discontinuous Galerkin methods, volume 69 of Mathematics and Applications. Springer, Heidelberg, 2012. doi:10.1007/978-3-642-22980-0
  • J. Droniou. Finite volume schemes for fully non-linear elliptic equations in divergence form. Math. Model. Numer. Anal., 40(6):1069–1100 (2007), 2006. doi:10.1051/m2an:2007001
  • J. Droniou. Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Meth. Appl. Sci. 24(8):1575–1619, 2014. doi:10.1142/S0218202514400041
  • J. Droniou and R. Eymard. A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105(1):35–71, 2006. doi:10.1007/s00211-006-0034-1
  • J. Droniou and R. Eymard. Study of the mixed finite volume method for Stokes and Navier–Stokes equations. Numer. Meth. Part. D. E. 25(1):137–171, 2009. doi:10.1002/num.20333
  • J. Droniou and R. Eymard. Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations. Numer. Math., 2015. doi:10.1007/s00211-015-0733-6
  • J. Droniou, R. Eymard, and P. Feron. Gradient schemes for Stokes problem. IMA J. Numer. Anal., 2015. doi:10.1093/imanum/drv061
  • J. Droniou, R. Eymard, T. Gallouet, C. Guichard, and R. Herbin. Gradient schemes for elliptic and parabolic problems. Unpublished.
  • J. Droniou, R. Eymard, T. Gallouet, and R. Herbin. A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Meth. Appl. Sci., 20(2):265–295, 2010. doi:10.1142/S0218202510004222
  • J. Droniou, R. Eymard, T. Gallouet, and R. Herbin. Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Meth. Appl. Sci. 23(13):2395–2432, 2012. doi:10.1142/S0218202513500358
  • J. Droniou, R. Eymard, and C. Guichard. Uniform-in-time convergence of numerical schemes for Richards' and Stefan's models. In M. Ohlberger J. Fuhrmann and C. Rohde Eds., editors, Finite Volumes for Complex Applications VII–-Methods and Theoretical Aspects, volume 77, pages 247–254. Springer, 2014. doi:10.1007/978-3-319-05684-5_23
  • J. Droniou, R. Eymard, and R. Herbin. Gradient schemes: generic tools for the numerical analysis of diffusion equations. Math. Model. Numer. Anal., 2015. doi:10.1051/m2an/2015079
  • J. Droniou, T. Gallouet, and R. Herbin. A finite volume scheme for a noncoercive elliptic equation with measure data. SIAM J. Numer. Anal., 41(6):1997–2031, 2003. doi:10.1137/S0036142902405205
  • J. Droniou, A. Porretta, and A. Prignet. Parabolic capacity and soft measures for nonlinear equations. Potential Anal., 19(2):99–161, 2003. doi:10.1023/A:1023248531928
  • J. Droniou and K. S. Talbot. On a miscible displacement model in porous media flow with measure data. SIAM J. Math. Anal., 46(5):3158–3175, 2014. doi:10.1137/130949294
  • R. Eymard, P. Feron, T. Gallouet, C. Guichard and R. Herbin. Gradient schemes for the Stefan problem. Int. J. Finite Vol., 10:1–37, 2013. http://www.i2m.univ-amu.fr/IJFV/spip.php?article47
  • R. Eymard, T. Gallouet, and R. Herbin. Finite volume methods. In P. G. Ciarlet and J.-L. Lions, editors, Techniques of Scientific Computing, Part III, Handbook of Numerical Analysis, VII, pages 713–1020. Elsevier, Amsterdam, 2000. https://www.elsevier.com/books/handbook-of-numerical-analysis/ciarlet/978-0-444-89928-6
  • R. Eymard, T. Gallouet, and R. Herbin. Cell centred discretisation of non linear elliptic problems on general multidimensional polyhedral grids. J. Numer. Math., 17(3):173–193, 2009. doi:10.1515/JNUM.2009.010
  • R. Eymard, T. Gallouet, and R. Herbin. Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal., 30(4):1009–1043, 2010. doi:10.1093/imanum/drn084
  • R. Eymard, C. Guichard, and R. Herbin. Small-stencil 3D schemes for diffusive flows in porous media. Math. Model. Numer. Anal. 46(2):265–290, 2012. doi:10.1051/m2an/2011040
  • R. Eymard, C. Guichard, R. Herbin, and R. Masson. Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation. J. Appl. Math. Mech., 94(7-8):560–585, 2014. doi:10.1002/zamm.201200206
  • R. Eymard, A. Handlovicova, R. Herbin, K. Mikula, and O. Stasova. Gradient schemes for image processing. In Finite volumes for complex applications VI Problems and Perspectives, volume 4 of Springer Proc. Math., pages 429–437. Springer, Heidelberg, 2011. doi:10.1007/978-3-642-20671-9_45
  • P. Fabrie and T. Gallouet. Modelling wells in porous media flow. Math. Models Meth. Appl. Sci., 10(5):673–709, 2000. doi:10.1142/S0218202500000367
  • I. Faille. Modelisation bidimensionnelle de la genese et de la migration des hydrocarbures dans un bassin sedimentaire. PhD thesis, Universite Joseph Fourier, Grenoble 1, 1992.
  • X. B. Feng. On existence and uniqueness results for a coupled system modeling miscible displacement in porous media. J. Math. Anal. Appl. 194(3):883–910, 1995. doi:10.1006/jmaa.1995.1334
  • T. Gallouet and R. Herbin. Finite volume approximation of elliptic problems with irregular data. In Finite volumes for complex applications II, pages 155–162. Hermes Sciences, Paris, 1999.
  • T. Gallouet and J.-C. Latche. Compactness of discrete approximate solutions to parabolic PDEs–-application to a turbulence model. Commun. Pure Appl. Anal. 11(6):2371–2391, 2012. doi:10.3934/cpaa.2012.11.2371
  • R. Glowinski and J. Rappaz. Approximation of a nonlinear elliptic problem arising in a non-newtonian fluid flow model in glaciology. Math. Model. Numer. Anal. 37(1):175–186, 2003. doi:10.1051/m2an:2003012
  • J. Simon. Compact sets in the space \(L^p(0,T;B)\). Annali Mat. Pura appl. 146:65–96, 1987. doi:10.1007/BF01762360

Published

2015-12-09

Issue

Section

Proceedings Computational Techniques and Applications Conference