In this work, local mesh adaptivity for the time harmonic Maxwell equations is studied. The main purpose is to apply a known a posteriori residual-based error estimator from the literature and to investigate its performance for a Y-beam splitter setting. This configuration is an important prototype for the design of optical systems within the excellence cluster PhoenixD. Specifically, the branching region is of interest and requires a high accuracy of the numerical simulation. One numerical example shows the performance of our approach

Beuchler, Sven

Kinnewig, Sebastian

König, Philipp

Wick, Thomas

English

Institutionelles Repositorium der Leibniz Universität Hannover

Received: 24 June 2021 Accepted: 20 September 2021DOI: 10.1002/pamm.202100175A residual-based error estimator and mesh adaptivity for the timeharmonic Maxwell equations applied to a Y-beam splitterSven Beuchler1,2, Sebastian Kinnewig1,2,∗, Philipp König1,2, and Thomas Wick1,21 Leibniz University Hannover, Institute of Applied Mathematics, Welfengarten 1, 30167 Hannover, Germany2 Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering - Innovation Across Disciplines), Leibniz UniversitätHannover, GermanyIn this work, local mesh adaptivity for the time harmonic Maxwell equations is studied. The main purpose is to apply aknown a posteriori residual-based error estimator from the literature and to investigate its performance for a Y-beam splittersetting. This configuration is an important prototype for the design of optical systems within the excellence cluster PhoenixD.Specifically, the branching region is of interest and requires a high accuracy of the numerical simulation. One numericalexample shows the performance of our approach.© 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.1 IntroductionThe Maxwell equations are well known for describing electromagnetic phenomena [5, 6]. One field of current interest areoptical technologies and the design of novel optical systems as for instance Y -splitters [8]. However, the branching regionof the Y -splitter requires in numerical approximations a high accuracy of the solution. For cost-complexity reasons, error-controlled local mesh adaptivity is a tool at hand. In this work, we concentrate on a re-implementation of a residual-basederror estimator presented in [4]; for early rigorous work we refer also to [9]. To this end, for the discretization Nédélec finiteelements are employed, e.g., [7], and the numerical solution strategy is based on a domain decomposition method [3] with ourown results provided in [2]. The implementation is done in the open-source finite element software deal.II [1].2 Time harmonic Maxwell equationsLet Ω ⊂ Rd, d ∈ {2, 3} be a bounded domain with the sufficiently smooth boundary Γ = Γ∞∪Γinc. Find E ∈ H (curl, Ω) :={v ∈ L2(Ω), curl (v) ∈ L2(Ω)} such thatcurl(µ−1curl (E))− ω2E = 0 in Ωµ−1γt (curl (E))− iωγT (E) = 0 on Γ∞γT (E) = γT(Einc)on Γinc, (1)where Einc : Rd → Cd, d ∈ {2, 3} is the incident field. Furthermore we define the traces γt (v) = n × v and γT (v) =n×(v×n) where n denotes the normal to Ω. We define ω = ϵ 2πλ , where λ > 0 is the wave length, ϵ is the relative permittivityand µ is the relative permeability. For a more detailed description see [2, 7].3 Discretization and adaptive mesh refinementProblem (1) is discretized by means of the h-version of the finite element method with Nedelec elements of a fixed polynomialdegree p = 2 on adaptively refined quadrilateral/hexehedral meshes with hanging nodes. To compute the error between thecontinuous solution E and the finite element approximation Eh, we concentrate on a residual-based strategy, estimating aglobal-norm error of the form ∥E − Eh∥. The resulting residual-based a posteriori error estimator is denoted by η(Eh) andwhich can be localized to single mesh elements K of the governing triangulation. Following [4], we specific form of localizedindicators readsηK(Eh)2 = ηR,K(Eh)2 + ηJ,K(Eh)2. (2)As usual, these indicators consists of two parts: an element-based residual term and face terms, respectively:ηR,K(Eh)2 :=h2Kp2(∥∥curl(µ−1curl (Eh))− ω2Eh − s∥∥2L2(K)+∥∥div(ω2Eh)∥∥2L2(K)),ηJ,K(Eh)2 :=12∑f ∈Fhfp(∥∥[γT(µ−1curl (Eh))]∥∥2L2(f)+∥∥[nf ·(ω2Eh + s)]∥∥2L2(f)),∗ Corresponding author: kinnewig@ifam.uni-hannover.deThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distributionin any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.PAMM · Proc. Appl. Math. Mech. 2021;21:1 e202100175. www.gamm-proceedings.com 1 of 2https://doi.org/10.1002/pamm.202100175 © 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH.2 of 2 Section 18: Numerical methods of differential equationswhere K denotes the current element with diameter hK and f denotes element faces with diameter hf . Moreover, nf denotesthe normal vector of f and s denotes the right hand side of the weak form.4 Numerical resultsIn this section, we realize the previously presented a posteriori error estimator using a Y -beam splitter configuration. There-fore, we consider the Y -beam splitter shown in figure 1, which is made out of a material with the refrective index n = 1.4906and is surrounded by air nair = 1.0 and the wavelength λ = 660 nm was used.Fig. 1: On the left side, the intensity plot of the x−y plane of the Y -beam splitter is shown. There, the red lines mark the interfaces betweenthe different domains. On the right side, the intensity at the output is visualized. The surrounding air is displayed in dark color.One major benefit of the combination of domain decomposition and adaptive grid refinement is that the first iteration stepsof the domain decomposition method can be done on a coarse grid and only in the last steps, the mesh will be refinenedadaptively, therefore speeding up the computations. As hypothesized, specifically the splitting region is adaptivitely refined,which is however driven automatically by the residual-based error estimator.Acknowledgements This work is funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy withinthe Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453).Open access funding enabled and organized by Projekt DEAL.References[1] D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin, D. Wells; Computers& Mathematics with Applications, Vol. 81, pp. 407-422, 2021[2] S. Beuchler, S. Kinnewig and T. Wick; arXiv:2105.11993v1, 2021[3] M. El Bouajaji, B. Thierry, X. Thierry and C. Geuzaine; Journal of Computational Physics, 2015, pp. 38-57[4] M. Bürg; Applied Numerical Mathematics, Vol. 72, 2013, pp. 188-204[5] R. P. Feynman; California Institute of Technology, Michael A. Gottlieb and Rudolf Pfeiffer, 1963[6] J. Maxwell; Oxford University Press, 1873[7] P. Monk; Oxford Science Publications, 2003.[8] K. Pflieger, L. Overmeyer; Lecture Notes in Networks and Systems, Vol 95, Springer, 2020[9] J. Schöberl; Math. Comp., Vol. 77 (262), 2008, pp. 633-649© 2021 The Authors. Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH. www.gamm-proceedings.com 16177061, 2021, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/pamm.202100175 by Technische Informationsbibliothek, Wiley Online Library on [17/10/2023]. 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A residual‐based error estimator and mesh adaptivity for the time harmonic Maxwell equations applied to a Y‐beam splitter

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A residual‐based error estimator and mesh adaptivity for the time harmonic Maxwell equations applied to a Y‐beam splitter

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