Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T15:52:19.282Z Has data issue: false hasContentIssue false
  • English
  • Français

On Modifications of Continuous and Discrete Maximum Principles for Reaction-Diffusion Problems

Published online by Cambridge University Press:  03 June 2015

István Faragó ,
Sergey Korotov  and
Tamás Szabó
István Faragó*
Affiliation:
Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117, Budapest, Pázmány P. s. 1/c, Hungary
Sergey Korotov*
Affiliation:
Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117, Budapest, Pázmány P. s. 1/c, Hungary BCAM - Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500, E-48160 Derio, Basque Country, Spain
Tamás Szabó*
Affiliation:
Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117, Budapest, Pázmány P. s. 1/c, Hungary
*
Email: faragois@cs.elte.hu
Corresponding author. Email: smkorotov@gmail.com, korotov@bcamath.org
Email: szabot@cs.elte.hu
Get access

Abstract

In this work, we present and discuss some modifications, in the form of two-sided estimation (and also for arbitrary source functions instead of usual sign-conditions), of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods.

Keywords

35B5065N0665N3065N50Reaction-diffusion problemmaximum principlediscrete maximum principlemonotone matrixtwo-sided a priori estimationcode validification
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 3 , Issue 1 , February 2011 , pp. 109 - 120
Copyright
Copyright © Global-Science Press 2011

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] Ahlberg, J. H. and Nilson, E. N., Convergence properties of the spline fit, J. SIAM., 11 (1963), pp. 95–104.Google Scholar
[2] Axelsson, O. and Kolotilina, L., Monotonicity and discretization error estimates, SIAM J. Numer. Anal., 27 (1990), pp. 1591–1611.Google Scholar
[3] Bramble, J. H. and Hubbard, B. E., On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type, J. Math. Phys., 43 (1964), pp. 117–132.Google Scholar
[4] Brandts, J., Korotov, S. and Křížek, M., The discrete maximum principle for linear sim-plicial finite element approximations of a reaction-diffusion problem, Linear. Algebra. Appl., 429 (2008), pp. 2344–2357.Google Scholar
[5] Brandts, J., Korotov, S., Křížek, M. and Šolc, J., On nonobtuse simplicial partitions, SIAM Rev., 51 (2009), pp. 317–335.Google Scholar
[6] Ciarlet, P. G., Discrete maximum principle for finite-difference operators, Aequationes. Math., 4 (1970), pp. 338–352.Google Scholar
[7] Ciarlet, P. G. and Raviart, P.-A., Maximum principle and uniform convergence for the finite element method, Comput. Methods. Appl. Mech. Engrg., 2 (1973), pp. 17–31.Google Scholar
[8] Faragó, I. and Horváth, R., Continuous and discrete parabolic operators and their qualitative properties, IMA Numer. Anal., 29 (2009), pp. 606–631.Google Scholar
[9] Forsythe, G. E. and Wasow, W. R., Finite-Difference Methods for Partial Differential Equations, Reprint of the 1960 original, Dover Phoenix Editions, Dover Publications, Inc., Mineola, NY, 2004.Google Scholar
[10] Hannukainen, A., Korotov, S. and Vejchodský, T., On weakening conditions for discrete maximum principles for linear finite element schemes, In “Proc. of the Fourth Internat. Conf. on Numerical Analysis and Applications (NAA–2008), Rousse, Bulgaria” (ed. by Margenov, S. et al.), LNCS 5434, Springer-Verlag, 2009, pp. 297–304.Google Scholar
[11] Hannukainen, A., Korotov, S. and Vejchodský, T., Discrete maximum principle for FE solutions of the diffusion-reaction problem on prismatic meshes, J. Comput. Appl. Math., 226 (2009), pp. 275–287.Google Scholar
[12] Hlaváček, I. and Křížek, M., On exact results in the finite element method, Appl. Math., 46 (2001), pp. 467–478.Google Scholar
[13] Karátson, J. and Korotov, S., Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numer. Math., 99 (2005), pp. 669–698.Google Scholar
[14] Korotov, S., Křížek, M. and Neittaanmäki, P., Weakened acute type condition for tetra-hedral triangulations and the discrete maximum principle, Math. Comp., 70 (2001), pp. 107–119.Google Scholar
[15] Korotov, S. and Vejchodský, T., A comparison of simplicial and block finite elements, In “Proc. of The Eighth Europ. Conf. on Numerical Mathematics and Advanced Applications (Enumath–2009), Uppsala, Sweden”, (ed. by Kreiss, G. et al.), Springer-Verlag, 2010, pp. 1–8 (in press).Google Scholar
[16] Křížek, M. and Lin, Qun, On diagonal dominance of stiffness matrices in 3D, East-West J. Numer. Math., 3 (1995), pp. 59–69.Google Scholar
[17] Křížek, M. and Neittaanmäki, P., Mathematical and Numerical Modelling in Electrical Engineering, Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
[18] Ladyzhenskaya, O. A. and Ural’tseva, N. N., Linear and Quasilinear Elliptic Equations, Leon Ehrenpreis Academic Press, New York-London, 1968.Google Scholar
[19] Lorenz, J., Zur inversmonotonie diskreter probleme, Numer. Math., 27 (1976/77), pp. 227– 238.Google Scholar
[20] Rózsa, P., Linear Algebra and Its Applications, Muszaki Konyvkiado, 1976 (in Hungarian).Google Scholar
[21] Smelov, V. V., Extension of the algebraic aspect of the discrete maximum principle, Russian J. Numer. Anal. Math. Model., 22 (2007), pp. 601–614.Google Scholar
[22] Varah, J. M., A lower bound for the smallest singular value of a matrix, Linear. Algebra. Appl., 11 (1975), pp. 3–5.Google Scholar
[23] Varga, R., Matrix Iterative Analysis, Prentice Hall, 1962.Google Scholar
[24] Varga, R., On discrete maximum principle, J. SIAM Numer. Anal., 3 (1966), pp. 355–359.Google Scholar
[25] Vejchodský, T. and Šolín, P., Discrete maximum principle for higher-order finite elements in 1D, Math. Comp., 76 (2007), pp. 1833–1846.Google Scholar
[26] Windisch, G., A maximum principle for systems with diagonally dominant M-matrices, in: Discretization in Differential Equations and Enclosures (ed. Adams, E. et al.), Math. Res., 36, Akademie-Verlag, Berlin, 1987, pp. 243–250.Google Scholar