Generalized Difference Methods for Differential Equations

Li, Ronghua; Chen, Zhongying; Wu, Wei

doi:10.1201/9781482270211

Cited by 261 publications

(241 citation statements)

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“…The basic idea of the finite volume method for general elliptic problems is to use the divergence theorem on the elliptic operator L of (1.1) to convert the double integral into a boundary integral as in (1.17). If one discretizes the boundary integral in (1.17) using finite differences, one gets the so-called finite volume difference methods [1,22] or the generalized difference methods [15,16,17]. On the other hand if one uses finite element spaces in the discretization, one gets the so-called finite volume element methods [3,4].…”

Section: Introductionmentioning

confidence: 99%

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Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach

1999

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Abstract. In this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane. We introduce a general approach for connecting these methods with finite element method analysis. This unified approach is used to prove known convergence results in the H 1 , L 2 norms and new results in the max-norm. For the elliptic problems we demonstrate that the error u − u h between the exact solution u and the approximate solution u h in the maximum norm is O(h 2 | ln h|) in the linear element case. Furthermore, the maximum norm error in the gradient is shown to be of first order. Similar results hold for the parabolic problems.

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Section: Introductionmentioning

confidence: 99%

“…160].) The approximation problem (1.14) has been considered by [16,17] where convergence results in the H 1 and L 2 norms were demonstrated. However, we shall prove these results in a unified way.…”

Section: Introductionmentioning

confidence: 99%

Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach

1999

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“…Finite volume element methods (FVEM) [6], also known as marker and cell methods, generalized difference methods [24], finite volume methods [23,32], covolume methods [7] or box methods [3,11], are approximation methods that could be placed somehow in between classical finite volume schemes and standard finite element (FE) methods. Roughly speaking, the FVEM is able to keep the simplicity and conservativity of finite volume methods and at the same time permits a natural development of error analysis in the L 2 −norm as in standard FE methods.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a finite volume element method for the Stokes problem

Quarteroni

Ruiz–Baier

2011

Numer. Math.

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Abstract. In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing projection method. The stabilization of the continuous lowest equal order pair finite volume element discretization (P 1 − P 1 ) is achieved by enriching the velocity space with bubble-like functions. Finally, some numerical experiments that confirm the predicted behavior of the method are provided.

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“…Here p * and p * are positive constants. Finite volume element (FVE) methods [3,6,7,16,17], also named as generalized difference methods [13,14,19,22] or box methods [1,11], have been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer or petroleum engineering. The FVE methods involve two spaces.…”

Section: Introductionmentioning

confidence: 99%

A second-order finite volume element method on quadrilateral meshes for elliptic equations

Yang¹

2006

ESAIM: M2AN

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Abstract. In this paper, by use of affine biquadratic elements, we construct and analyze a finite volume element scheme for elliptic equations on quadrilateral meshes. The scheme is shown to be of second-order in H 1 -norm, provided that each quadrilateral in partition is almost a parallelogram. Numerical experiments are presented to confirm the usefulness and efficiency of the method.Mathematics Subject Classification. 65N30, 65N15.

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Generalized Difference Methods for Differential Equations

Cited by 261 publications

References 0 publications

Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach

Error estimates in $L^2$, $H^1$ and $L^\infty$ in covolume methods for elliptic and parabolic problems: A unified approach

Analysis of a finite volume element method for the Stokes problem

A second-order finite volume element method on quadrilateral meshes for elliptic equations

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