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

Acosta, Gabriel; Armentano, María

doi:10.1090/S0025-5718-2011-02481-6

Finite element approximations in a non-Lipschitz domain: Part II
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by Gabriel Acosta and María G. Armentano PDF

Math. Comp. 80 (2011), 1949-1978 Request permission

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 $\Omega \subset \mathbb {R}^2$, 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 $\Omega _h$ verifies $\Omega \subset \Omega _h$, in the above-mentioned paper the source term of the Poisson problem was taken equal to $0$ outside $\Omega$ 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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