A Compact Difference Scheme for the Biharmonic Equation in Planar Irregular Domains

Ben‐Artzi, Matania; Chorev, I.; Croisille, Jean‐Pierre; Fishelov, Dalia

doi:10.1137/080718784

Cited by 34 publications

(29 citation statements)

References 27 publications

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“…Discontinuous Galerkin methods have also been recently developed and analysed [20,23,28,19]; error estimates have been derived for polynomials of degree greater or equal to two or three. Other methods which have been developed for fourth order problems include mixed methods [6] (see also references therein), [29], and compact finite difference methods [7,3,2].…”

Section: Introductionmentioning

confidence: 99%

Finite volume schemes for the biharmonic problem on general meshes

Eymard

Gallouët²,

Herbin³

et al. 2012

Math. Comp.

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During the development of a convergence theory for Nicolaides' extension [21,24] of the classical MAC scheme [25,22,26] for the incompressible Navier-Stokes equations to unstructured triangle meshes, it became clear that a convergence theory for a new kind of finite volume discretizations for the biharmonic problem would be a very useful tool in the convergence analysis of the generalized MAC scheme. Therefore, we present and analyze new finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions on grids which satisfy an orthogonality condition. We prove that a piece-wise constant approximate solution of the biharmonic problem converges in L 2 (Ω) to the exact solution. Similar results are shown for the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. Error estimates are also derived. This part of the paper is a first, significant step towards a convergence theory of Nicolaides' extension of the classical MAC scheme. Further, we show that finite volume discretizations for the biharmonic problem can also be defined on very general, nonconforming meshes, such that the same convergence results hold. The possibility to construct a converging lowest order finite volume method for the H 2 -regular biharmonic problem on general meshes seems to be an interesting result for itself and clarifies the necessary ingredients for converging discretizations of the biharmonic problem. All these results are confirmed by numerical results.

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Section: Introductionmentioning

confidence: 99%

Finite volume schemes for the biharmonic problem on general meshes

Eymard

Gallouët²,

Herbin³

et al. 2012

Math. Comp.

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“…The second approach consists with the splitting mechanism where the biharmonic equation is splitted into two coupled Poisson equations (see [17], and the references therein) such that one of these Poisson equations has no boundary conditions. Very recently, significant contributions have been made to solve N-S equations using compact biharmonic formulation on nonuniform grids [4,33]. The application of compact schemes to the biharmonic form of N-S equations is fairly recent.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Implicit Compact Streamfunction Velocity Formulation of Two Dimensional Flows

Pandit

Karmakar

2016

J Sci Comput

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In this paper, we develop a class of implicit, unconditionally stable compact schemes for solving coupled fourth order and second order semilinear streamfunction and temperature equations respectively representing unsteady Navier-Stokes (N-S) equations for incompressible viscous fluid flows. The governing equations are presented in a generic form from which specific equations are obtained for several two dimensional (2D) incompressible viscous fluid flow problems. The streamfunction and the temperature equations are all solved iteratively in a coupled system of equations for the four field variables consisting of streamfunction, two velocities and temperature. The discretized form of biharmonic equation in streamfunction consists on 5-point stencil of streamfunction and is found to be second-order accurate both for spatially and temporally. In addition, we have considered two strategies for the discretization of the energy equation, which are second order accurate 5-point and fourth order accurate 9-point compact scheme. The formulation is made efficient by decreasing the computational complexity and is used to solve several 2D time dependent well studied benchmark problems. It is seen to efficiently capture both time wise and steady-state solutions of the N-S equations with Dirichlet as well as Neumann boundary conditions. The results obtained using our proposed scheme are in excellent agreement with the results available in the literature and they clearly establish the efficiency and the accuracy of the proposed scheme.

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“…1 Work supported by Groupement de recherche MOMAS, PACEN/CNRS. Throughout this paper, d ∈ N \ {0} denotes the space dimension, Ω is an open polygonal bounded and connected subset of R d , with Lipschitz-continuous boundary ∂Ω, and f ∈ L 2 (Ω), ∈ L 2 (Ω) and g ∈ (L 2 (Ω)) d .…”

Section: Introductionmentioning

confidence: 99%

Approximation of the biharmonic problem using piecewise linear finite elements

Eymard

Herbin²

2010

Comptes Rendus. Mathématique

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We propose an approximation of the solution of the biharmonic problem in H 2 0 (Ω) which relies on the discretization of the Laplace operator using nonconforming continuous piecewise linear finite elements. r é s u m éNous proposons une approximation de la solution du problème bi-harmonique dans H 2 0 (Ω) basée sur la discrétisation du Laplacien par éléments finis P1 continus mais non conformes.

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A Compact Difference Scheme for the Biharmonic Equation in Planar Irregular Domains

Cited by 34 publications

References 27 publications

Finite volume schemes for the biharmonic problem on general meshes

Finite volume schemes for the biharmonic problem on general meshes

An Efficient Implicit Compact Streamfunction Velocity Formulation of Two Dimensional Flows

Approximation of the biharmonic problem using piecewise linear finite elements

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