Elliptic problems in a non Lipschtiz domain

MARIA GABRIELA ARMENTANO

UXBRIDGE - Londres

Congreso; MAFELAP 2016; 2016

Brunel University

In this work we review and analyze the approximation, by standard piecewise linear finite elements, of some elliptic problems in the plane domain $\Omega =\{(x; y) : 0 < x < 1; 0 < y < x^{\alpha}\}$; which gives, for $\alpha >1 $, the simplest model of an external cusp.The focus of  interest resides in the fact that, since the domain is  curved and non-Lipschitz,  the problems  under consideration had not  been covered by the standard  literature which only had dealt with polygonal or smooth domains.First, since many of the results on Sobolev spaces, which are fundamental in the usual error analysis, do not apply to cusp domains  [G], we had to develop  trace  and  extension theorems  in weighted Sobolev spaces, with the weight being a power of the distance to the cuspidal.  These estimates  allowed  us  to prove, for the Poisson problem,that the optimal order, with respect to the number of nodes, could be recovered by using appropriate graded meshes [AADL1, AADL2, AA1].Then, we  studied the Laplacian eigenvalue problem, in which the classical spectral theory could not be applied directly, and in consequence, this eigenvalue problem had to be  reformulated in a proper setting in order to obtain quasi optimal order of convergence for the eigenpairs [AA2]. At  present,  we are studying  a Steklov eigenvalue problem and the particular difficulties that arise in this problem.Bibliography[AA1] G. Acosta and M. G. Armentano (2011), Finite element approximations in a non-Lipschitz domain: Part. II, Math. Comp.  80 (276), pp. 1949-1978 .[AA2] G. Acosta and M. G. Armentano (2014), Eigenvalue Problem in a non-Lipschitz domain, IMA Journal of Numerical Analysis.  34 (1), pp. 83-95.[AADL1]  G. Acosta, M. G. Armentano, R. G. Durán and A. L. Lombardi (2005), Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp, Journal of Mathematical Analysis and Applications  310(2), pp. 397-411.[AADL2]  G. Acosta, M. G. Armentano, R. G. Durán and A. L. Lombardi (2007), Finite element approximations in a non-Lipschitz  domain, SIAM J. Numer. Anal.  45 (1), pp. 277-295.[G] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.