FoCM 2014 conference
Workshop C2 - Foundation of Numerical PDE's - Semi-plenary talk
December 18, 17:05 ~ 17:55 - Room A22
Finite element spectral approximation of the curl operator in multiply connected domains
Rodolfo Rodriguez
Departamento de Ingenieria Matematica, Universidad de Concepcion, Chile - rodolfo@ing-mat.udec.cl
A couple of numerical methods based on Nedelec finite elements have been recently introduced and analyzed in [1] to solve the eigenvalue problem for the curl operator in simply connected domains. This topological assumption is not just a technicality, since the eigenvalue problem is ill-posed on multiply connected domains, in the sense that its spectrum is the whole complex plane as is shown in [2]. However, additional constraints can be added in order to recover a well posed problem with a discrete spectrum [2,3]. We choose as additional constraint a zero-flux condition of the curl on all the cutting surfaces. We introduce two weak formulations of the corresponding problem, which are convenient variations of those studied in [1]; one of them is mixed and the other a Maxwell-like formulation. We prove that both are well posed and show how to modify the finite element discretization from [1] to take care of these additional constraints. We prove spectral convergence of both discretizations and establish a priori error estimates. We also report numerical tests which allow assessing the performance of the proposed methods.
[1] R. Rodriguez and P. Venegas, Numerical approximation of the spectrum of the curl operator, Math. Comp. (online: S 0025-5718(2013)02745-7).
[2] Z. Yoshida and Y. Giga, Remarks on spectra of operator rot, Math. Z., 204 (1990) 235--245.
[3] R. Hiptmair, P.R. Kotiuga and S. Tordeux, Self-adjoint curl operators, Ann. Mat. Pura Appl., 191 (2012) 431--457.