A family of monotone methods for the numerical solution of three-dimensional diffusion problems on unstructured tetrahedral meshes

Kapyrin, Ivan

doi:10.1134/s1064562407050249

Cited by 43 publications

(34 citation statements)

References 3 publications

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“…If one manages to write all the convex combinations (16) using only cell centers, then (26) involves boundary values only if K is a boundary cell (i.e., A K ∩A ext = ∅); in this case, for f ≥ 0 and u ∂ = 0, the solution to the scheme cannot attain is minimum on an interior cell unless it is constant.…”

Section: Fig 3 Illustration Of Assumption 26 (Choice Of M Ia In Tmentioning

confidence: 99%

“…Study of the scheme. Assumption 2.6 being satisfied, we denote by S the scheme (27), where the interior fluxes are given by (26) and the boundary fluxes are given by (18), with the notation (16), (17), (21), (23), and (25) and the choice (24).…”

Section: Fig 3 Illustration Of Assumption 26 (Choice Of M Ia In Tmentioning

confidence: 99%

“…In [30], a cell-centered finite volume discretization for diffusion operators is proposed, and its robustness and accuracy are shown through comparisons with analytical solutions. This scheme satisfies either the minimum or the maximum principle but not both principles simultaneously; it has been extended in [26,34,35,38] on polygonal meshes and tetrahedrons. However, no scheme developed in these articles satisfies with certainty both the minimum and the maximum principles simultaneously, and no theoretical proof of convergence is given.…”

Section: Introduction Let ω Be An Open Bounded Connected Polygonal Dmentioning

confidence: 99%

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Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle

Droniou¹,

Potier²

2011

SIAM J. Numer. Anal.

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Abstract. We present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models.

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Section: Fig 3 Illustration Of Assumption 26 (Choice Of M Ia In Tmentioning

confidence: 99%

Section: Fig 3 Illustration Of Assumption 26 (Choice Of M Ia In Tmentioning

confidence: 99%

Section: Introduction Let ω Be An Open Bounded Connected Polygonal Dmentioning

confidence: 99%

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Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle

Droniou¹,

Potier²

2011

SIAM J. Numer. Anal.

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“…The value of ϕ ′ (µ) is updated by formula (21). The values of µ generated in this way correspond to the jog points.…”

Section: Implementation Issuesmentioning

confidence: 99%

“…To guarantee solution monotonicity for arbitrary meshes, a number of nonlinear methods have been proposed in both FE [12,31] and finite volume [4,16,21,24,25,27,28,29,33,38,39] frameworks. We present here a procedure for postprocessing non-monotone FE solution which produces a corrected solution satisfying both monotonicity and DMP requirements, and also preserving the order of accuracy.…”

Section: Introductionmentioning

confidence: 99%

Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions

Burdakov

Kapyrin

Vassilevski

2012

Journal of Computational Physics

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We suggest here a least-change correction to available finite element (FE) solution. This postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may not be exhibited by the FE solution. It is based on solving a monotonic regression problem with some extra constraints. One of them is a linear equality-type constraint which models the conservativity requirement. The other ones are box-type constraints, and they originate from the discrete maximum principle. The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessed FE solution preserves the accuracy of the discrete FE approximation.We introduce an algorithm for solving the postprocessing problem. It can be viewed as a dual ascent method based on the Lagrangian relaxation of the equality constraint. We justify theoretically its correctness. Its efficiency is demonstrated by the presented results of numerical experiments.

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A monotone finite volume scheme for advection–diffusion equations on distorted meshes

Wang

Yuan

et al. 2011

Numerical Methods in Fluids

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SUMMARY A new monotone finite volume method with second‐order accuracy is presented for the steady‐state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes that guarantee the positivity of the numerical solution. The approximation of the diffusive flux is based on nonlinear two‐point approximation, and the approximation of the advective flux is based on the second‐order upwind method with proper slope limiter. The second‐order convergence rate for concentration and the monotonicity of the nonlinear finite volume method are verified with numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.

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A family of monotone methods for the numerical solution of three-dimensional diffusion problems on unstructured tetrahedral meshes

Cited by 43 publications

References 3 publications

Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle

Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle

Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions

A monotone finite volume scheme for advection–diffusion equations on distorted meshes

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