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Numerische Mathematik

, Volume 127, Issue 3, pp 515–537 | Cite as

Maximum principle in linear finite element approximations of anisotropic diffusion–convection–reaction problems

  • Changna LuEmail author
  • Weizhang Huang
  • Jianxian Qiu
Article

Abstract

A mesh condition is developed for linear finite element approximations of anisotropic diffusion–convection–reaction problems to satisfy a discrete maximum principle. Loosely speaking, the condition requires that the mesh be simplicial and \(\mathcal {O}(\Vert \varvec{b}\Vert _\infty h + \Vert c\Vert _\infty h^2)\)-nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix, where \(h\) denotes the mesh size and \(\varvec{b}\) and \(c\) are the coefficients of the convection and reaction terms. In two dimensions, the condition can be replaced by a weaker mesh condition (an \(\mathcal {O}(\Vert \varvec{b}\Vert _\infty h + \Vert c\Vert _\infty h^2)\) perturbation of a generalized Delaunay condition). These results include many existing mesh conditions as special cases. Numerical results are presented to verify the theoretical findings.

Mathematics Subject Classification

65N30 65N50 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.College of Math and StatisticsNanjing University of Information Science and TechnologyNanjingChina
  2. 2.Department of MathematicsThe University of KansasLawrenceUSA
  3. 3.School of Mathematical SciencesXiamen UniversityXiamenChina

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