Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Transparent Boundary Conditions

Abstract

The time-harmonic Maxwell equations at high wavenumber k are discretized by edge elements of degree p on a mesh of width h. For the case of a ball as the computational domain and exact, transparent boundary conditions, we show quasi-optimality of the Galerkin method under the k-explicit scale resolution condition that a) kh/p is sufficient small and b) p/ log k is bounded from below. *

Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems

“…• a bounded analytic domain with impedance boundary conditions (Melenk & Sauter, 2011, Case I), or • a 2D polygonal domain with impedance boundary conditions (Melenk & Sauter, 2011, Case II) and (Esterhazy & Melenk, 2012, Equation 9), or • the exterior of an analytic domain with Dirichlet boundary conditions on the scatterer and the Sommerfeld radiation condition at infinity (Melenk & Sauter, 2011, Case III). We also mention similar results for the Maxwell system set in a ball with transparent boundary conditions, see Melenk & Sauter (2018).…”

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“…This observation implies that the above operators admit so-called "element-by-element" constructions: Remark 2.9. As described in more detail in [26,Sec. 8] , Π L 2 p , that map into the standard spaces of piecewise polynomials W p+1 (T ), Nédélec spaces Q p (T ), or Raviart-Thomas spaces V p (T ).…”

Wavenumber Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems

“…5] and in particular the proof of[15, Thm. 5.5] for details.Similarly as in[14], one can show that for the Galerkin method based on S := S p,1 (\scrT h ) and fixed constant C > 0, there exists a constant c > 0 such that the resolution condition Downloaded 11/06/20 to 130.60.47.196. Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms ensures convergence and quasi-optimality.…”