scholarly journals Integral equations via saddle point problems for time-harmonic Maxwell's equations

2003 ◽  
Vol 150 (1) ◽  
pp. 157-192 ◽  
Author(s):  
Francis Collino ◽  
Bruno Despres
2018 ◽  
Vol 18 (3) ◽  
pp. 451-475 ◽  
Author(s):  
Alexandre Ern ◽  
Jean-Luc Guermond

AbstractWe devise a novel framework for the error analysis of finite element approximations to low-regularity solutions in nonconforming settings where the discrete trial and test spaces are not subspaces of their exact counterparts. The key is to use face-to-cell extension operators so as to give a weak meaning to the normal or tangential trace on each mesh face individually for vector fields with minimal regularity and then to prove the consistency of this new formulation by means of some recently-derived mollification operators that commute with the usual derivative operators. We illustrate the technique on Nitsche’s boundary penalty method applied to a scalar diffusion equation and to the time-harmonic Maxwell’s equations. In both cases, the error estimates are robust in the case of heterogeneous material properties. We also revisit the error analysis framework proposed by Gudi where a trimming operator is introduced to map discrete test functions into conforming test functions. This technique also gives error estimates for minimal regularity solutions, but the constants depend on the material properties through contrast factors.


Author(s):  
Rainer Mandel ◽  
Robert Schippa

AbstractWe solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.


Sign in / Sign up

Export Citation Format

Share Document