Finite element approximation of convection diffusion problems using graded meshes

Durán, Ricardo G.; Lombardi, Ariel L.

doi:10.1016/j.apnum.2006.03.029

Cited by 44 publications

(54 citation statements)

References 2 publications

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“…Recently Durán and Lombardi [6] presented a kind of graded meshes for the convection dominated convectiondiffusion problems. This mesh can be regarded as an improvement of Shishkin-type meshes and do not need any transition point in essential.…”

Section: Introductionmentioning

confidence: 99%

Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

Zhu¹,

Chen²

2008

Journal of Computational and Applied Mathematics

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The bilinear finite element methods on appropriately graded meshes are considered both for solving singular and semisingular perturbation problems. In each case, the quasi-optimal order error estimates are proved in the -weighted H 1 -norm uniformly in singular perturbation parameter , up to a logarithmic factor. By using the interpolation postprocessing technique, the global superconvergent error estimates in -weighted H 1 -norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.

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Section: Introductionmentioning

confidence: 99%

Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

Zhu¹,

Chen²

2008

Journal of Computational and Applied Mathematics

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“…To overcome this difficulty, we turn to the DG method based on a lay-adapted mesh. Because this mesh is different from the grade mesh proposed by R. G. Duran and L. Lombardi in [14], we denote it by improved grade mesh. The detailed construction of improved grade meshes is defined below.…”

Section: Theorem 32mentioning

confidence: 98%

A robust discontinuous Galerkin method for solving convection-diffusion problems

Zhang

Xie

Tao

2008

Acta Math. Appl. Sin. Engl. Ser.

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In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed.Particularly, the 2p + 1-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically.

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“…In [24,10], the authors introduce the so-called graded meshes: for a given h>0, and a constant >0, the partition…”

Section: Graded Meshes and A Priori Estimatesmentioning

confidence: 99%

“…Furthermore, as far as we know, constructing such kind of piecewise quasi-uniform meshes needs a transition point, which is not easy to determine. Recently, a kind of graded meshes was presented in [10] for the convection-dominated convection-diffusion problems. These graded meshes can be regarded as an improvement of Shishkin-type meshes, and it has a characteristic that there are comparable sizes between the adjacent intervals, which makes it more robust than Shishkin-type meshes.…”

mentioning

confidence: 99%

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Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems

Zhu

Chen

2008

Math Methods in App Sciences

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SUMMARYThe numerical approximation by a lower-order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving semisingular perturbation problems. The quasi-optimal-order error estimates are proved in the ε-weighted H 1 -norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H 1 -norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.

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Finite element approximation of convection diffusion problems using graded meshes

Cited by 44 publications

References 2 publications

Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

A robust discontinuous Galerkin method for solving convection-diffusion problems

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for semisingularly perturbed reaction–diffusion problems

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