The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges

Apel, Thomas; Nicaise, Serge

doi:10.1002/(sici)1099-1476(199804)21:6<519::aid-mma962>3.0.co;2-r

Cited by 71 publications

(78 citation statements)

References 19 publications

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“…This regularity will be used to define the meshes, in Section 3, in order to be able to estimate the H 1 -seminorm of the linear Lagrange interpolation error for a solution of this problem, for values of p close to 2, in a satisfactory way. We follow the article [1]. Let S be a corner of Ω.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Under this condition, uniform and anisotropic error estimates have been obtained for different interpolation operators. In particular, we will use the results obtained for Lagrange interpolation in [1] and for Nédélec interpolation in [12].…”

Section: The Maximum Angle Conditionmentioning

confidence: 99%

“…More precisely, our meshes are proposed in order to be able to adequately approximate a homogeneous Dirichlet problem for the Laplace operator with a right hand side in L p for some p > 2. These meshes were designed in [1]. So, the results contained in that article become fundamental for our approach.…”

Section: Introductionmentioning

confidence: 99%

“…• Accurate error estimates for a continuous piecewise polynomial interpolation of the H 1 0 (Ω)-solution of the scalar Laplace equation with right hand side in L p [1].…”

Section: Introductionmentioning

confidence: 99%

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The discrete compactness property for anisotropic edge elements on polyhedral domains

Lombardi

2012

ESAIM: M2AN

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Section: Notation and Preliminariesmentioning

confidence: 99%

Section: The Maximum Angle Conditionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…• Accurate error estimates for a continuous piecewise polynomial interpolation of the H 1 0 (Ω)-solution of the scalar Laplace equation with right hand side in L p [1].…”

Section: Introductionmentioning

confidence: 99%

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The discrete compactness property for anisotropic edge elements on polyhedral domains

Lombardi

2012

ESAIM: M2AN

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“…As pointed out in [26], quasi-uniform meshes are weakly quasi-uniform. Moreover graded meshes of Raugel's type used in the presence of corner and edge singularities (see for instance [2]) are also weakly quasi-uniform.…”

Section: Discontinuous Galerkin Discretizationmentioning

confidence: 99%

Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system

Creusé¹,

Nicaise²

2006

ESAIM: M2AN

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Abstract. In this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's system on quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtained and the convergence of the discrete eigenvalues to the continuous ones is deduced using the theory of collectively compact operators. Some numerical experiments confirm the theoretical predictions.Mathematics Subject Classification. 65N25, 65N30.

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Interpolation inh‐Version Finite Element Spaces

Apel

2004

Encyclopedia of Computational Mechanics

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Interpolation operators associate with a function an element from a finite element space. The difference between the two functions is called interpolation error . Estimates of the interpolation error are used for a priori and a posteriori estimation of the discretization error of finite element methods. For a priori error estimates, one can often use the nodal interpolant . To get optimal error bounds, the maximum available regularity of the solution has to be used. The situation is different for a posteriori error estimates. For the investigation of residual‐type error estimators, local interpolation error estimates for functions from the Sobolev space W 1, 2 (Ω) are needed. Such estimates can be obtained for many finite elements only for quasi‐interpolation operators , where point values of functions or derivatives are replaced in the definition of the interpolation operators by certain averaged values. This paper gives an overview of different interpolation operators and their error estimates. The discussion restricts the h ‐version of the finite element method, but it includes interpolation on the basis of triangular/tetrahedral and quadrilateral/hexahedral meshes, affine and nonaffine elements, isotropic and anisotropic elements, and Lagrangian and other elements.

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The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges

Cited by 71 publications

References 19 publications

The discrete compactness property for anisotropic edge elements on polyhedral domains

The discrete compactness property for anisotropic edge elements on polyhedral domains

Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system

Interpolation inh‐Version Finite Element Spaces

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