Finite Element Approximations in a NonLipschitz Domain

In this paper we analyze the approximation by standard piecewise linear finite elements of a non homogeneous Neumann problem in a cuspidal domain. Since the domain is not Lipschitz, many of the results on Sobolev spaces which are fundamental in the usual error analysis do not apply. Therefore, we need to work with weighted Sobolev spaces and to develop some new theorems on traces and extensions. We show that, in the domain considered here, suboptimal order can be obtained with quasiuniform meshes even when the exact solution is in H, and we prove that the optimal order with respect to the number of nodes can be recovered by using appropriate graded meshes.