Finite element approximation   of spectral problems   with Neumann  boundary  conditions   on curved domains

Hernández, Erwin; Rodrı́guez, Rodolfo

doi:10.1090/s0025-5718-02-01467-9

Cited by 14 publications

(22 citation statements)

References 22 publications

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“…Moreover, since Ω = Ω h , the standard theory for eigenvalue approximations [9] does not apply straightforwardly. In fact, the study of convergence for this problems leads to problems such as (1.4) with f not necessarily equal to zero outside Ω [17,24]. On the other hand, the study of the error between a certain extension of the solution u and u h , analyzed in Section 5, is also of interest in the context of eigenvalue approximations [17,24].…”

Section: Figure 1 Cuspidal Domainmentioning

confidence: 99%

“…On the other hand, in eigenvalue approximations the usual approach (see [9]) is based on the convergence of appropriate operators T h to the limit operator T , with T being the inverse of the Laplacian. Since Ω h = Ω, the operators T h are mesh dependent and the analysis leads to the study of problems such as (1.4) with f not necessarily equal to zero outside Ω [17].…”

Section: Error Estimates In the Case In Which Fmentioning

confidence: 99%

“…The results given below allow us, in particular, to obtain a precise computation of terms like u h L 2 (Ω h \Ω) which, for example, provides an optimal bound for the error between u L 2 (Ω) and u h L 2 (Ω h ) . On the other hand, estimates for the error between u E and u h are useful in the analysis of the error of eigenvalue problems [17].…”

Section: From (12) and (14) We Getmentioning

confidence: 99%

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Finite element approximations in a non-Lipschitz domain: Part II

Acosta

Armentano

2011

Math. Comp.

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Abstract. In a paper by R. Durán, A. Lombardi, and the authors (2007) the finite element method was applied to a non-homogeneous Neumann problem on a cuspidal domain Ω ⊂ R 2 , and quasi-optimal order error estimates in the energy norm were obtained for certain graded meshes. In this paper, we study the error in the L 2 norm obtaining similar results by using graded meshes of the type considered in that paper. Since many classical results in the theory Sobolev spaces do not apply to the domain under consideration, our estimates require a particular duality treatment working on appropriate weighted spaces.On the other hand, since the discrete domain Ω h verifies Ω ⊂ Ω h , in the above-mentioned paper the source term of the Poisson problem was taken equal to 0 outside Ω in the variational discrete formulation. In this article we also consider the case in which this condition does not hold and obtain more general estimates, which can be useful in different problems, for instance in the study of the effect of numerical integration, or in eigenvalue approximations.

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Section: Figure 1 Cuspidal Domainmentioning

confidence: 99%

Section: Error Estimates In the Case In Which Fmentioning

confidence: 99%

Section: From (12) and (14) We Getmentioning

confidence: 99%

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Finite element approximations in a non-Lipschitz domain: Part II

Acosta

Armentano

2011

Math. Comp.

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“…These papers deal only with Dirichlet boundary conditions. For Neumann boundary conditions the first results have been obtained even more recently ( [25]). …”

Section: Introductionmentioning

confidence: 98%

Finite element approximation of spectral acoustic problems on curved domains

Hernández

Rodrı́guez²

2004

Numerische Mathematik

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This paper deals with the finite element approximation of the displacement formulation of the spectral acoustic problem on a curved non convex two-dimensional domain . Convergence and error estimates are proved for Raviart-Thomas elements on a discrete polygonal domain h ⊂ in the framework of the abstract spectral approximation theory. Similar results have been previously proved only for polygonal domains. Numerical tests confirming the theoretical results are reported. (2000): 65N25, 65N30, 70J30 Mathematics Subject Classification

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“…The finite element approximation of eigenvalues and eigenfunctions on curved domains is considered in a great number of papers [5][6][7][11][12][13]1]. The interest rests on their significant practical relevance.…”

Section: Introductionmentioning

confidence: 99%

Finite element approximation of the elasticity spectral problem on curved domains

Hernández

2009

Journal of Computational and Applied Mathematics

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a b s t r a c tWe analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non-convex domain. In the framework of the abstract spectral approximation theory, we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors. Two kinds of problems are considered: the discrete domain does not coincide with the real one and mixed boundary conditions are imposed. Some numerical results are presented.

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Finite element approximation of spectral problems with Neumann boundary conditions on curved domains

Cited by 14 publications

References 22 publications

Finite element approximations in a non-Lipschitz domain: Part II

Finite element approximations in a non-Lipschitz domain: Part II

Finite element approximation of spectral acoustic problems on curved domains

Finite element approximation of the elasticity spectral problem on curved domains

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