Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
Perugia, Ilaria
Trefftz finite element methods are schemes whose approximating spaces are locally made by solutions of the partial differential equation to be approximated. In the case of wave problems in frequency domain, Trefftz spaces contain oscillating functions with the same frequency as the original problem. The resulting methods feature enhanced convergence properties with respect to standard polynomial finite elements. Prominent examples of Trefftz-type methods are the ultra weak variational formulation (UWVF) by Cessenat and Després, the partition of unit finite element method (PUFEM) by Babuška and Melenk, the discontinuous enrichment method (DEM/DGM) by Farhat and co-workers, the variational theory of complex rays (VTCR) by Ladevèze, and the wave based method (WBM) by Desmet. par In this talk, we focus on a family of Trefftz-discontinuous Galerkin (TDG) methods, which includes the UWVF as a special case, for the approximation of the time-harmonic Maxwell equation [ nablatimes(mu^-1nablatimesmathbfE) - k^2varepsilonmathbfE=mathbf0 ] in a bounded domain, with impedance boundary conditions. The presented TDG method is unconditionally well-posed and quasi-optimal a in a mesh-dependent energy-type norm, i.e., well-posedness and convergence are proved for any value of the wave number and of the mesh size. By a modified duality argument, error estimates in a mesh-independent norm, which is slightly weaker than $L^2$, are also derived. This analysis has required to develop new stability estimates and regularity results for the continuous problem which might be of interest on their own. The particular case where the approximating Trefftz spaces are made of vector-valued plane waves is considered, and explicit error estimates are derived. Finally, some hints on the extension of these methods to the case of scattering problems are given. par These results have been obtained in collaboration with Ralf Hiptmair (ETH Zürich) and Andrea Moiola (University of Reading).