FINITE ELEMENT APPROXIMATIONS IN A NON-LIPSCHITZ DOMAIN: PART II

GABRIEL ACOSTA, MARÍA G. ARMENTANO

MATHEMATICS OF COMPUTATION

AMER MATHEMATICAL SOC

Año: 2011 vol. 80 p. 1949 - 1978

0025-5718

In [2] the finite element method was applied to a non-homogeneous Neumann prob-lem on a cuspidal domain ­ $Omega subset  R^2$, 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 [2]. 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 ­$Omega_h$ verfies  $Omega \subset Omega_h$, in [2] the source term of the Poisson problem was taken equal to 0 outside ­ 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 diferent problems, for instance in the study of the efect of numerical integration, or in eigenvalue approximations.