Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle

Korotov, Sergey; ížek, Michal Kř; Neittaanmäkia, Pekka

doi:10.1090/s0025-5718-00-01270-9

Cited by 106 publications

(84 citation statements)

References 16 publications

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“…In Ruas Santos [4] examples were given of triangulations of non-Delauney type, for which the elliptic maximum principle still holds. In R 3 , Korotov, Křížek, and Neittaanmäki [3] showed an elliptic maximum principle for tetrahedral decompositions of nonacute type. (For certain convection dominated elliptic cases, cf.…”

Section: Introductionmentioning

confidence: 99%

On the existence of maximum principles in parabolic finite element equations

Thomée

Wahlbin

2008

Math. Comp.

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Abstract. In 1973, H. Fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space. In particular, he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute, and this is known to generalize in two space dimensions to triangulations of Delauney type. In this note we consider more general parabolic equations and first show that a maximum principle cannot hold for the standard spatially semidiscrete problem. We then show that for the lumped mass method the above conditions on the triangulation are essentially sharp. This is in contrast to the elliptic case in which the requirements are weaker. We also study conditions for the solution operator acting on the discrete initial data, with homogeneous lateral boundary conditions, to be a contraction or a positive operator.

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Section: Introductionmentioning

confidence: 99%

On the existence of maximum principles in parabolic finite element equations

Thomée

Wahlbin

2008

Math. Comp.

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“…They were used by various authors to prove the convergence of the lowest-order finite difference and finite element methods (see, e.g., [3,4] and the references therein). DMP have been studied intensively during the past decades in the context of linear PDEs [2,8,10,17,18,20] and more recently also nonlinear equations [9]. Most of these results have two points in common:…”

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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“…A more relaxed version of Condition 1.2 has been obtained in [7]; it allows for the existence of slightly negative edges (the sum of the opposite angles has to be bounded by π + for some > 0) but adds restrictions involving a larger neighborhood of each edge. This result has been extended to three dimensions in [5].…”

Section: ∇U · ∇Vmentioning

confidence: 79%

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

2004

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Abstract. There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green's function is positive for triangular meshes allowing sufficiently good approximation of H 1 functions. We study this question for the Poisson problem in two dimensions discretized via the Galerkin method with continuous piecewise linears. We give examples which show that in general the answer is negative, and furthermore we extend the number of cases where it is known to be positive. Our techniques utilize some new results about discrete Green's functions that are of independent interest.

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Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle

Cited by 106 publications

References 16 publications

On the existence of maximum principles in parabolic finite element equations

On the existence of maximum principles in parabolic finite element equations

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

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

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