Finite element approximations in a non Lipschitz domain

MARÍA G. ARMENTANO

Workshop; Alfa-Meeting; 2005

The finite element element method has been widely analyzed in itsdifferent forms for all kind of partial differential equations.However, as far as we know, all the analysis are restricted to thecase of polygonal or smooth domains and no results have beenobtained for the case in which the domain is not Lipschitz.In this work we obtain optimal order error estimates for linearfinite element approximations of the solution of a nonhomogeneousNeumann problem for the Poisson equation in $Omega$, a planedomain  with an external cusp, by using appropriate graded meshes.In order to approximate the solution by finite element methods,the simplest method is to replace $Omega$ by a polygonal domain$Omega_h$ with vertices on the boundary of $Omega$ and apply thestandard linear finite element method. In a recent paper we provedthat, under appropriate assumptions on the boundary data, thesolution of the problem belongs to $H^2(Omega)$ so one wouldexpect to obtain optimal order of convergence with this approachby using quasi-uniform meshes. However, numerical examples showthat this approximation may not be optimal. The reason for thisbehavior seems to be the fact that although $u in H^2(Omega)$ itcan not be extended to a function in $H^2(Omega_h)$ indeed, inthis kind of domains the standard extension theorem does notapply.First, we present an extension theorem for the domain $O$ whichshows that the solution of our problem  can be extended to afunction in a weighted $H^2$ space where the weight is a power ofthe distance to the cuspidal point. Then, we prove that takingappropriate graded meshes the linear finite element methodconverges in $H^1$ with optimal order respect of the number ofnodes. We conclude the work by presenting several numericalexamples.