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

J. Karátson, S. Korotov

Research output: Contribution to journalArticle

64 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)669-698
Number of pages30
JournalNumerische Mathematik
Volume99
Issue number4
DOIs
Publication statusPublished - Feb 2005

Fingerprint

Discrete Maximum Principle
Nonlinear Elliptic Problems
Maximum principle
Mixed Boundary Conditions
Finite Element Solution
Boundary conditions
Maximum Principle
Preservation
Mesh
Qualitative Properties
Linear Approximation
Finite Element Approximation
Piecewise Linear
Acute
Mathematical models
Scalar
Mathematical Model
Computer simulation
Numerical Simulation
Sufficient Conditions

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. / Karátson, J.; Korotov, S.

In: Numerische Mathematik, Vol. 99, No. 4, 02.2005, p. 669-698.

Research output: Contribution to journalArticle

@article{0e2201dfffc04925827cf4d0599e86a5,
title = "Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions",
abstract = "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.",
author = "J. Kar{\'a}tson and S. Korotov",
year = "2005",
month = "2",
doi = "10.1007/s00211-004-0559-0",
language = "English",
volume = "99",
pages = "669--698",
journal = "Numerische Mathematik",
issn = "0029-599X",
publisher = "Springer New York",
number = "4",

}

TY - JOUR

T1 - Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions

AU - Karátson, J.

AU - Korotov, S.

PY - 2005/2

Y1 - 2005/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=14544271374&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=14544271374&partnerID=8YFLogxK

U2 - 10.1007/s00211-004-0559-0

DO - 10.1007/s00211-004-0559-0

M3 - Article

AN - SCOPUS:14544271374

VL - 99

SP - 669

EP - 698

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 4

ER -