A monotone finite element scheme for convection-diffusion equations

Xu, Jinchao; Zikatanov, Ludmil T.

doi:10.1090/s0025-5718-99-01148-5

Cited by 221 publications

(237 citation statements)

References 31 publications

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“…(For certain convection dominated elliptic cases, cf. [7] and references therein.) Thus the parabolic maximum principle demands more stringent conditions than (1.13).…”

Section: Introductionmentioning

confidence: 99%

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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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“…(For certain convection dominated elliptic cases, cf. [7] and references therein.) Thus the parabolic maximum principle demands more stringent conditions than (1.13).…”

Section: Introductionmentioning

confidence: 99%

“…In [2] the condition used was that each of these angles is ≤ π/2, and, for higher space dimensions, similar conditions of "acute" type are used. In Xu and Zikatanov [7] sharp conditions of Delauney type were given for S −1 to be nonpositive in any number of space dimensions.…”

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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“…Nonetheless, specialized numerical schemes tailored for this class of non-coercive problems could easily be used instead. We mention for instance monotone and stabilized FEM, finite volume methods (FVM) or combined FEM-FVM, see [20,23,42,55].…”

Section: Introductionmentioning

confidence: 99%

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Abdulle

Huber

2017

Numer. Math.

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We introduce a numerical homogenization method based on a discontinuous Galerkin finite element heterogeneous multiscale method (DG-HMM) to efficiently approximate the effective solution of parabolic advection-diffusion problems with rapidly varying coefficients, large Péclet number and compressible flows. To estimate the missing data of an effective model, numerical upscaling is performed which accurately captures the effects of microscopic solenoidal or gradient flow at a macroscopic scale such as enhancement or depletion of the effective diffusion. For compressible flow with periodic data, we derive sharp a priori error estimates for the macro and micro discretization errors which are robust in the advection dominated regime. Numerical tests confirm the error estimates for problems with periodic data and illustrate the applicability of our method for problems with non-periodic data.

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A monotone finite element scheme for convection-diffusion equations

Cited by 221 publications

References 31 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

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

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