A nonconforming combination  of the finite element and volume methods  with an anisotropic mesh refinement  for a singularly perturbed  convection-diffusion equation

Wang, Song; Li, Zi‐Cai

doi:10.1090/s0025-5718-03-01516-3

Cited by 9 publications

(12 citation statements)

References 21 publications

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“…We mention, however, that combined FE-FV discretizations of (1.5) have been considered also under assumption (1.6). For example, error estimates are derived in [22] for a pairing of nonconforming linear triangular finite elements with exponentially fitted finite volumes on a mixture of triangular and rectangular grids. Mixed finite volume schemes are considered in [23] (stability and error estimates).…”

Section: Introductionmentioning

confidence: 99%

Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

Deuring¹,

Mildner²

2010

Numerical Methods Partial

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We consider a time-dependent and a stationary convection-diffusion equation. These equations are approximated by a combined finite element -finite volume method: the diffusion term is discretized by CrouzeixRaviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the nonstationary case, we use an implicit Euler approach for time discretization. This scheme is shown to be L 2 -stable uniformly with respect to the diffusion coefficient. In addition, it turns out that stability is unconditional in the time-dependent case. These results hold if the underlying grid satisfies a condition that is fulfilled, for example, by some structured meshes.

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Section: Introductionmentioning

confidence: 99%

Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

Deuring¹,

Mildner²

2010

Numerical Methods Partial

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“…Sufficient conditions for the existence of this decomposition have been discussed in many literatures [6,11,15,17,22,25]. The following lemma shows that v and all its first and second partial derivatives are uniformly bounded in Ω 1 .…”

Section: The Galerkin Finite Element Formulationmentioning

confidence: 95%

“…To avoid non-physical numerical solutions, many special finite element techniques have been developed, including upwind finite element [1,4], Petrov-Galerkin finite element [7], streamline diffusion finite element methods [2,8,9], and exponentially fitted finite elements [18,[21][22][23]. However, these methods do not always give accurate results, especially when a diffusion coefficient has the same magnitude as that of mesh size.…”

Section: Introductionmentioning

confidence: 99%

Error Estimates of the Finite Element Method with Weighted Basis Functions for a Singularly Perturbed Convection-Diffusion Equation

Li¹,

Chen

2011

JCM

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In this paper, we establish a convergence theory for a finite element method with weighted basis functions for solving singularly perturbed convection-diffusion equations. The stability of this finite element method is proved and an upper bound O(h| ln ε| 3/2 ) for errors in the approximate solutions in the energy norm is obtained on the triangular Bakhvalov-type mesh. Numerical results are presented to verify the stability and the convergent rate of this finite element method.Mathematics subject classification: 65N30, 35J20.

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“…There exist many reports on the study of singularly perturbed differential equations such as [2][3][4][5][6][7] to just name a few. In this paper, we follow our recent papers [8,1], and use the technique of separation of variables (cf., for example, [9]) to seek new and better test problems involving singularly perturbed differential equations.…”

Section: Introductionmentioning

confidence: 99%

Accurate and approximate analytic solutions of singularly perturbed differential equations with two-dimensional boundary layers

Tsai

Wang

et al. 2008

Computers & Mathematics with Applications

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In this paper we construct three new test problems, called Models A, B and C, whose solutions have two-dimensional boundary layers. Approximate analytic solutions are found for these problems, which converge rapidly as the number of terms in their expansion increases. The approximations are valid for = 10 −8 in practical computations. Surprisingly, the algorithm for Model A can be carried out even for → ∞. Model C has a simple exact solution. These three new accurate and approximate analytic solutions with two-dimensional boundary layers may be more useful for testing numerical methods than those in [Z.Particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains, I. Convergence analysis, J. Comput. Appl. Math. 166 (2004) 181-208] in the sense that the series solutions from the former converge much faster than those of the latter when is small.

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A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

Cited by 9 publications

References 21 publications

Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

Error Estimates of the Finite Element Method with Weighted Basis Functions for a Singularly Perturbed Convection-Diffusion Equation

Accurate and approximate analytic solutions of singularly perturbed differential equations with two-dimensional boundary layers

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