On discrete maximum principles for nonlinear elliptic problems

Authors:
János Karátson;Sergey Korotov;Michal Kříek
Affiliations:
Department of Applied Analysis, Eötvös Loránd University, H-1518 Budapest, Pf. 120, Hungary;Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FI-02015 TKK, Finland;Institute of Mathematics, Academy of Sciences, itná 25, CZ-115 67 Prague 1, Czech Republic
Venue:
Mathematics and Computers in Simulation
Year:
2007

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Abstract

In order to have reliable numerical simulations it is very important to preserve basic qualitative properties of solutions of mathematical models by computed approximations. For scalar second-order elliptic equations, one of such properties is the maximum principle. In our work, we give a short review of the most important results devoted to discrete counterparts of the maximum principle (called discrete maximum principles, DMPs), mainly in the framework of the finite element method, and also present our own recent results on DMPs for a class of second-order nonlinear elliptic problems with mixed boundary conditions.