Error Estimates for Finite Element Approximations in Domains with an External Cusp

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

UADE- Buenos Aires, Argentina

Congreso; MECOM 2005; 2005

In this work we obtain optimal order error estimates for linear finite element approximations of the solution of a nonhomogeneous Neumann problem for the Poisson equation in $Omega$, a plane domain  with an external cusp, by using appropriate graded meshes. In [1] we proved that, under appropriate assumptions on the boundary data, the solution of the problem belongs to $H^2(Omega)$. 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 the standard linear finite element with a quasi-uniform mesh. In smooth domains one would expect to obtain optimal order of convergence with this approach, however, for cuspidal domains this approximation may not be optimal. The reason for this behavior seems to be the fact that although $u in H^2(Omega)$ it can not be extended to a function in $H^2(Omega_h)$ indeed, in this kind of domains the standard extension theorem does not apply (see [3]). In this work we define an extension of the solution which lies in a weighted Sobolev space and using it we prove that taking appropriate graded meshes, of the type considered in [2], the linear finite element method converges in $H^1$ with optimal orderrespect of the number of nodes. We conclude the work by presenting several numerical examples. References[1] G. Acosta, M. G. Armentano, R. G. Durán and A. Lombardi, Nonhomogeneous Neumann problem for the Poisson equation in  domains with an external cusp, to appear in Journal ofMathematical Analysis and Applications (2005). [2] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, (1985) [3] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, (1970)