A weak discrete maximum principle for hp-FEM
- 1 December 2007
- journal article
- Published by Elsevier BV in Journal of Computational and Applied Mathematics
- Vol. 209 (1), 54-65
- https://doi.org/10.1016/j.cam.2006.10.028
Abstract
In this paper, we prove a new discrete maximum principle (DMP) for the one-dimensional Poisson equation discretized by the hp-FEM. While the DMP for piecewise-linear elements is a classical result from the 1970s, no extensions to hp-FEM are available to the present day. Due to a negative result by Höhn and Mittelmann from 1981, related to quadratic Lagrange elements, it was long assumed that higher-order finite elements do not satisfy discrete maximum principles. In this paper we explain why it is not possible to make a straightforward extension of the classical DMP to the higher-order case, and we propose stronger assumptions on the right-hand side under which an extension is possible.Keywords
This publication has 10 references indexed in Scilit:
- Discrete conservation of nonnegativity for elliptic problems solved by the hp-FEMMathematics and Computers in Simulation, 2007
- Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditionsNumerische Mathematik, 2005
- On the maximum and comparison principles for a steady-state nonlinear heat conduction problemZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2003
- Goal-oriented error estimation and adaptivity for the finite element methodComputers & Mathematics with Applications, 2001
- Weakened acute type condition for tetrahedral triangulations and the discrete maximum principleMathematics of Computation, 2000
- Approximation properties of the h-p version of the finite element methodComputer Methods in Applied Mechanics and Engineering, 1996
- Toward a universal adaptive finite element strategy, part 1. Constrained approximation and data structureComputer Methods in Applied Mechanics and Engineering, 1989
- Some remarks on the discrete maximum-principle for finite elements of higher orderComputing, 1981
- Maximum principle and uniform convergence for the finite element methodComputer Methods in Applied Mechanics and Engineering, 1973
- Discrete maximum principle for finite-difference operatorsAequationes mathematicae, 1970