Failure of the discrete maximum principle   for an elliptic finite element problem

Drăgănescu, Andrei; Dupont, Todd; Scott, L. Ridgway

doi:10.1090/s0025-5718-04-01651-5

Cited by 91 publications

(81 citation statements)

References 15 publications

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“…The present result is based on the analysis of the discrete Green's function (DGF) for higher-order elements. A similar concept was used in the piecewise-linear case in [5]. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Discrete maximum principle for higher-order finite elements in 1D

Vejchodský¹,

Šolín

2007

Math. Comp.

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Abstract. We formulate a sufficient condition on the mesh under which we prove the discrete maximum principle (DMP) for the one-dimensional Poisson equation with Dirichlet boundary conditions discretized by the hp-FEM. The DMP holds if a relative length of every element K in the mesh is bounded by a value H * rel (p) ∈ [0.9, 1], where p ≥ 1 is the polynomial degree of the element K. The values H * rel (p) are calculated for 1 ≤ p ≤ 100.

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Section: Introductionmentioning

confidence: 99%

Discrete maximum principle for higher-order finite elements in 1D

Vejchodský¹,

Šolín

2007

Math. Comp.

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“…where C > 0 is independent of the meshsize h. In [8], positivity for discrete Green's function was investigated for Poisson equations. The authors addressed the question of whether the discrete Green's function is positive for triangular meshes allowing sufficiently good approximation of H 1 functions.…”

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confidence: 99%

Maximum Principles for $P1$-Conforming Finite Element Approximations of Quasi-linear Second Order Elliptic Equations

Wang¹,

Zhang²

2012

SIAM J. Numer. Anal.

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Abstract. This paper derives some discrete maximum principles for P 1-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial differential equations to finite element methods. The mathematical tools are based on the variational approach that was commonly used in the classical PDE theory. The discrete maximum principles are established by assuming a property on the discrete variational form that is of global nature. In particular, the assumption on the variational form is verified when the finite element partition satisfies some angle conditions. For the general quasi-linear elliptic equation, these angle conditions indicate that each triangle or tetrahedron needs to be O(h α )-acute in the sense that each angle α ij (for triangle) or interior dihedral angle α ij (for tetrahedron) must satisfy α ij ≤ π/2 − γh α for some α ≥ 0 and γ > 0. For the Poisson problem where the differential operator is given by Laplacian, the angle requirement is the same as the existing ones: either all the triangles are non-obtuse or each interior edge is non-negative. It should be pointed out that the analytical tools used in this paper are based on the powerful De Giorgi's iterative method that has played important roles in the theory of partial differential equations. The mathematical analysis itself is of independent interest in the finite element analysis.

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“…The discrete maximum principle is also of interest to avoid negative numerical values of typical positive physical quantities like concentration, temperature (in Kelvins), density, and pressure. A single obtuse triangle in the partition can destroy the discrete maximum principle [22,78] It can easily be shown that each nonpositive continuous function f = 0 whose support does not intersect the supports of the finite element functions that vanish at v 1 gives rise to an approximation u h of u that is positive at v 2 , hence violating the discrete maximum principle. It should be mentioned that an alternative approach to enforcing the validity of discrete maximum principles is to modify the finite element method in such a way that obtuse angles do not stand in their way anymore.…”

Section: Applications In Numerical Mathematicsmentioning

confidence: 99%

On Nonobtuse Simplicial Partitions

Brandts¹,

Korotov²,

Křížek³

et al. 2009

SIAM Rev.

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On nonobtuse simplicial partitionsBrandts, J.H.; Korotov, S.; Kížek, M.; Šolc, J. Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. Download date: 10 May 2018Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. Dedicated to Ivan Hlaváček on his 75th birthdayAbstract. This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

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Failure of the discrete maximum principle for an elliptic finite element problem

Cited by 91 publications

References 15 publications

Discrete maximum principle for higher-order finite elements in 1D

Discrete maximum principle for higher-order finite elements in 1D

Maximum Principles for $P1$-Conforming Finite Element Approximations of Quasi-linear Second Order Elliptic Equations

On Nonobtuse Simplicial Partitions

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