Abstract
When choosing a numerical method to approximate the solution of a continuous mathematical problem, we need to consider which method results in an approximation that is not only close to the solution of the original problem, but possesses the important qualitative properties of the original problem, too. For linear elliptic problems the main qualitative properties are the various maximum principles. The preservation of the weak maximum principle was extensively investigated in the last decades, but not the strong maximum principle preservation. In this paper we focus on the latter property by giving its necessary and sufficient conditions, investigating the relation of the preservation of the strong and weak maximum principles and illustrating the differences between them with numerous examples.
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
Ciarlet, P.G.: Discrete maximum principle for finite-difference operators. Aequationes Math. 4, 338–352 (1970)
Draganescu, A., Dupont, T.F., Scott, L.R.: Failure of the discrete maximum principle for an elliptic finite element problem. Math. Comp. 74(249), 1–23 (2005)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. AMS (1997)
Faragó, I.: Numerical Treatment of Linear Parabolic Problems. Dissertation for the degree MTA Doktora (2008)
Hannukainen, A., Korotov, S., Vejchodský, T.: On Weakening Conditions for Discrete Maximum Principles for Linear Finite Element Schemes. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2008. LNCS, vol. 5434, pp. 297–304. Springer, Heidelberg (2009)
Ishihara, K.: Strong and weak discrete maximum principles for matrices associated with elliptic problems. Linear Algebra Appl. 88-89, 431–448 (1987)
Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)
Varga, R.S.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1962)
Varga, R.: On discrete maximum principle. J. SIAM Numer. Anal. 3, 355–359 (1966)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mincsovics, M.E., Horváth, T.L. (2012). On the Differences of the Discrete Weak and Strong Maximum Principles for Elliptic Operators. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2011. Lecture Notes in Computer Science, vol 7116. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29843-1_70
Download citation
DOI: https://doi.org/10.1007/978-3-642-29843-1_70
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29842-4
Online ISBN: 978-3-642-29843-1
eBook Packages: Computer ScienceComputer Science (R0)