Domain Decomposition Methods for Time-Harmonic Electromagnetic Waves With High-Order Whitney Forms

Abstract

Classically, domain decomposition methods (DDM) for time-harmonic electromagnetic wave propagation problems make use of the standard, low order, Nédélec basis functions. This paper analyzes the convergence rate of DDM when higher order finite elements are used for both volume and interface discretizations, in particular when different orders are used in the volume and on the interfaces.

A Rigorous Code Verification Process of the Domain Decomposition Method in a Finite Element Method for Electromagnetics

“…Garcia-Castillo are with the Department of Signal Theory and Communications, University Carlos III of Madrid, 28911, Leganes, Spain (e-mail: aamor@ing.uc3m.es, legcasti@ing.uc3m.es) O. Floch, L.L. Toth, and R. Dyczij-Edlinger are with the Lehrstuhl für Theoretische Elektrotechnik, Universität des Saarlandes, Saarbrücken [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42]. All these references can be divided into three groups: i) optimized Schwarz methods, focused on the optimization of the so-called transmission conditions, [16], [25], [30]; ii) cement element methods, which use transmission conditions through the definition of cement variables with physical meaning, [13], [14], [17], [18], [19], [20], [21], [22]; iii) finite element tear and interconnecting (FETI) methods, which use Lagrange multipliers to obtain a reduced system of equations, [15], [23], [26], [27], [24], [32],…”

“…As an additional advantage, we can show how rigorous the accuracy of the method is since the convergence rates should follow the same pattern as in conventional FEM. A convergence study is rarely present when implementing DDM, as in [20], [22], [29]. Typically, the implementation of DDM is checked after postprocessing the field with the same codes without DDM, [15], with other commercial software, [37], with analytical solutions, [20] or even with measurements, [26].…”

Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities

“…To the best of our knowledge, those OS techniques share a common drawback: they ignore the impact of back-propagating waves. While this assumption is legitimate in many cases (antenna arrays [15], medical imaging reconstruction [16] or photonic waveguides [17] just to cite a few), it becomes questionable when the geometry allows resonances (even if the source does not oscillate exactly at a resonance frequency), as found for instance in lasers [18], accelerator cavities [19] or quantum electrodynamic devices [7].…”

Convergence of classical optimized non-overlapping Schwarz method for Helmholtz problems in closed domains

“…To the best of our knowledge, all OS techniques share a common drawback: they ignore the impact of backpropagating waves. While this assumption is legitimate in many cases (antenna arrays [14], medical imaging reconstruction [15] or photonic waveguides [16] just to cite a few), it becomes questionable when the geometry allows resonances (even if the source does not oscillate exactly at a resonance frequency), as found for instance in lasers [17], accelerator cavities [18] or quantum electrodynamic devices [7].…”