Finite element approximations in a non-Lipschitz domain

GABRIEL ACOSTA, MARÍA G. ARMENTANO, RICARDO G. DURÁN,

Londres, Inglaterra

Simposio; MAFELAP 2006; 2006

The Brunel Institute of Computational Mathematics

In this work we analyze the approximation, by standard piecewise linear finite elements, of a non homogeneous Neumann problem in a domain with an external cusp. Since the domain is not Lipschitz many of the results on Sobolev spaces, which are fundamental in the usual error analysis, do not apply. In a recent paper we proved that, under appropriate assumptions onthe boundary data, the solution of the problem belongs to $H^2$. One could think that, since the solution is regular, the numerical approximation obtained with quasi-uniform meshes would be of optimal order. However, numerical examples show that this is not the case. The reason for this behavior seems to be the fact that the solution can not be extended to an $H^2$ function on the polygonal domain approximating the original domain. Indeed, it is known that the standard extension theorems in Sobolev spaces do not apply for our domain. We prove that the optimal order, with respect to the number of nodes in the $H^1$ norm, can be recovered by using appropriate graded meshes.  To obtain this result, we first prove an extension theorem for our domain which shows that the solution of our problem  can be extended to a function in a eighted $H^2$ space, with the weight being a power of the distance to the cuspidal point.