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Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions

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Summary.

One of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations. For problems of elliptic type, one of the basic properties is the (continuous) maximum principle. In our work, we present several variants of the maximum principles and their discrete counterparts for (scalar) second-order nonlinear elliptic problems with mixed boundary conditions. The problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes. Sufficient conditions providing the validity of the corresponding discrete maximum principles are presented. Geometrically, they mean that the employed meshes have to be of acute or nonobtuse type, depending of the type of the problem. Finally some examples of real-life problems, where the preservation of maximum principles plays an important role, are presented.

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Authors and Affiliations

  1. Department of Applied Analysis, Eötvös Loránd University, 1518, Budapest, Pf. 120, Hungary

    J. Karátson

  2. Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35, 40014, Jyväskylä, Finland

    S. Korotov

Authors
  1. J. Karátson
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  2. S. Korotov
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Correspondence to S. Korotov.

Additional information

The first author was supported by the Hungarian Research Fund OTKA under grant no. F034840

The second author was supported by the Agora Center under Grant InBCT of TEKES, Finland, and by the Academy Research Fellowship no. 208628 from the Academy of Finland

Mathematics Subject Classification (2000): 35B50, 35J65, 65N30, 65N50

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Karátson, J., Korotov, S. Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math. 99, 669–698 (2005). https://doi.org/10.1007/s00211-004-0559-0

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