Summary: FINITE ELEMENT APPROXIMATIONS IN A NON-LIPSCHITZ DOMAIN
GABRIEL ACOSTA, MAR´IA G. ARMENTANO, RICARDO G. DUR´AN, AND ARIEL L. LOMBARDI
Abstract. 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 funda-
mental 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 quasi-
uniform meshes even when the exact solution is in H2
, and we prove that the optimal order
with respect to the number of nodes can be recovered by using appropriate graded meshes.
The finite element method has been widely analyzed in its different forms for all kind of
partial differential equations. However, as far as we know, all analyses are restricted to the case
of polygonal or smooth domains and no results have been obtained for the case in which the
domain is non Lipschitz, with the exception of the well known fracture problems.
The goal of this paper is to start the analysis of finite element approximations in non-Lipschitz
domains. As a first step in this direction we consider a model problem in a plane domain with
an external cusp.
Several difficulties arise in this problem because many of the results on Sobolev spaces, which