2024/08/08 by Chaumont-Frelet, Théophile, Galkowski, Jeffrey, Spence, Euan A. · 1 citation
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2408.04507
We prove sharp wavenumber-explicit error bounds for first- or second-type-Nédélec-element (a.k.a. edge-element) conforming discretisations, of arbitrary (fixed) order, of the variable-coefficient time-harmonic Maxwell equations posed in a bounded domain with perfect electric conductor (PEC) boundary conditions. The PDE coefficients are allowed to be piecewise regular and complex-valued; this set-up therefore includes scattering from a PEC obstacle and/or variable real-valued coefficients, with the radiation condition approximated by a perfectly matched layer (PML). In the analysis of the h-version of the finite-element method, with fixed polynomial degree p, applied to the time-harmonic Maxwell equations, the asymptotic regime is when the meshwidth, h, is small enough (in a wavenumber-dependent way) that the Galerkin solution is quasioptimal independently of the wavenumber, while the preasymptotic regime is the complement of the asymptotic regime. The results of this paper are the first preasymptotic error bounds for the time-harmonic Maxwell equations using first-type Nédélec elements or higher-than-lowest-order second-type Nédélec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.