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Research Group of Prof. Dr. Barbara Verfürth

Contact Information

Address:
Institut für Numerische Simulation
Friedrich-Hirzebruch-Allee 7
53115 Bonn
Phone: +49 228 73-69833
Office: FHA7 3.030
E-Mail: ed tod nnob-inu tod sni ta htreufreva tod b@foo tod de

News/Open Position

We have a job opening for a research assistant (PhD student or PostDoc). The candidate will work on the (numerical) homogenization of time-varying metamaterials, a project funded through the collaborative research center CRC1173 “Wave phenomena”. For further details, please contact Prof. Dr. Barbara Verfuerth stating why you are interested and sending a CV and transcripts.

Research interests

  • numerical methods for PDEs
  • multiscale (finite element) methods
  • numerical homogenization
  • (time-harmonic) wave propagation, Helmholtz and Maxwell equations
  • quasilinear PDEs

Teaching

Summer semester 2024

Winter semester 2023/24

Summer semester 2023

See teaching activities of the whole group.

Research Projects

Current

Homogenization of time-varying metamaterials

Project B4, DFG CRC 1173.

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Numerical methods for nonlinear, random and dynamical multiscale problems

Project 496556642, DFG Emmy Noether.

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Metamaterials are modern, artificially constructed materials that are tailored to exhibit new, astonishing physical properties. Therefore, they play a decisive role to control and manipulate waves, for instance in laser applications. Metamaterials are characterized by fine structures of different material components. The typical length of these fine structures is much smaller than the length of the whole material bulk. Further important building blocks in applications are nonlinear material responses and time modulation. Finally, the robustness of the material properties with respect to imperfections in the fabrication process is highly relevant. Within mathematical models, these applications lead to partial differential equations with a coexistence of multiple temporal and spatial scales, nonlinearities, and random perturbations. Numerical simulations have very high potential in the material design as they can replace time-consuming and costly experiments. Yet, standard numerical methods need to resolve all fine material structures so that their computational effort is prohibitive even with today’s computer resources. In contrast, computational multiscale methods (CMMs) deliver a macroscopic representation of the solution by suitable local upscaling processes. However, the incorporation of nonlinearities, random perturbations, and multiscale dynamics require new computational paradigms for several reasons. Firstly, CMMs often rely on linear arguments that break down for nonlinear problems. Thus, most approaches propose to couple nonlinear problems on fine and macroscopic scales in a rather complicated manner. Secondly, Monte Carlo techniques require many multiscale simulations with new, rather costly upscaling processes for each random sample. Present approaches at least for the numerical analysis additionally rely on stochastic homogenisation results. Thirdly, CMMs for dynamical problems mostly treat multiple spatial or multiple temporal scales exclusively.In this project, we develop and analyse novel CMMs to tackle nonlinear, randomly perturbed, and dynamical problems. The main goals are connected to fundamental mathematical and computational challenges requiring a revolutionary coalescence of multiscale methods, model reduction, uncertainty quantification, and time integration. We (a) explore adaptive linearised and nonlinear approximation spaces for nonlinear multiscale problems, (b) unite multiscale methods and Monte Carlo approaches for randomly perturbed problems, and (c) bridge spatial and temporal multiscale methods for rapid multiscale dynamics. While the general nature of our approaches allows their application to a wide range of problems, we put special emphasis on wave-related phenomena. Moreover, we rigorously justify all methods by error estimates which is crucial beyond the experimentally validated regime. Ultimately, this project will push forward the frontiers of CMMs for realistic (metamaterial) applications.

Completed

TEEMLEAP - A new TEstbed for Exploring Machine LEarning in Atmospheric Prediction

KIT Future Fields.

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See all projects of the group.

Publications

Preprints

  1. Statistical variational data assimilation. A. Benaceur and B. Verfürth. arXiv preprint 2305.04734, 2023. BibTeX arXiv
  2. Metamaterial applications of TMATSOLVER, an easy-to-use software for simulating multiple wave scattering in two dimensions. S. C. Hawkins, L. G. Bennetts, M. A. Nethercote, M. A. Peter, D. Peterseim, H. J. Putley, and B. Verürth. submitted, preprint avaialable upon reasonable request, 2023. BibTeX

Articles

  1. Wave propagation in high-contrast media: periodic and beyond. É. Fressart and B. Verfürth. Comp. Meth. Appl. Math., 2024. BibTeX DOI
  2. Two-step homogenization of spatiotemporal metasurfaces using an eigenmode-based approach. P. Garg, A. G. Lamprianidis, S. Rahman, N. Stefanou, E. Almpanis, N. Papanikolaou, B. Verfürth, and C. Rockstuhl. Opt. Mater. Express, 14(2):549–563, feb 2024. See supplement https://doi.org/10.6084/m9.figshare.24849822.v2. BibTeX DOI
  3. Higher-order finite element methods for the nonlinear Helmholtz equation. B. Verfürth. J. Sci. Comput., 2024. online first. BibTeX DOI arXiv
  4. Numerical multiscale methods for waves in high-contrast media. B. Verfürth. Jahresber. Dtsch. Math.-Ver., pages 37–65, 2024. BibTeX DOI
  5. Fully discrete heterogeneous multiscale method for parabolic problems with multiple spatial and temporal scales. D. Eckhardt and B. Verfürth. BIT, 63(2):35, 2023. BibTeX DOI arXiv
  6. Modeling four-dimensional metamaterials: a T-matrix approach to describe time-varying metasurfaces. P. Garg, A. G. Lamprianidis, D. Beutel, T. Karamanos, B. Verfürth, and C. Rockstuhl. Opt. Express, 30(25):45832–45847, 2022. BibTeX DOI preprint
  7. Numerical Upscaling for Wave Equations with Time-Dependent Multiscale Coefficients. B. Maier and B. Verfürth. Multiscale Model. Simul., 20(4):1169–1190, 2022. BibTeX DOI arXiv
  8. Multiscale scattering in nonlinear Kerr-type media. R. Maier and B. Verfürth. Math. Comp., 91(336):1655–1685, 2022. BibTeX DOI
  9. Nonlinear Helmholtz equations with sign-changing diffusion coefficient. R. Mandel, Z. Moitier, and B. Verfürth. C. R. Math. Acad. Sci. Paris, 360:513–538, 2022. BibTeX DOI
  10. An offline-online strategy for multiscale problems with random defects. A. Målqvist and B. Verfürth. ESAIM Math. Model. Numer. Anal., 56(1):237–260, 2022. BibTeX DOI
  11. Numerical homogenization for nonlinear strongly monotone problems. B. Verfürth. IMA J. Numer. Anal., 42(2):1313–1338, 2022. BibTeX DOI
  12. A multiscale method for heterogeneous bulk-surface coupling. R. Altmann and B. Verfürth. Multiscale Model. Simul., 19(1):374–400, 2021. BibTeX DOI
  13. A generalized finite element method for problems with sign-changing coefficients. T. Chaumont-Frelet and B. Verfürth. ESAIM Math. Model. Numer. Anal., 55(3):939–967, 2021. BibTeX DOI
  14. A diffuse modeling approach for embedded interfaces in linear elasticity. P. Hennig, R. Maier, D. Peterseim, D. Schillinger, B. Verfürth, and M. Kästner. GAMM-Mitt., 43(1):e202000001, 16, 2020. BibTeX DOI
  15. Mathematical analysis of transmission properties of electromagnetic meta-materials. M. Ohlberger, B. Schweizer, M. Urban, and B. Verfürth. Netw. Heterog. Media, 15(1):29–56, 2020. BibTeX DOI
  16. Computational high frequency scattering from high-contrast heterogeneous media. D. Peterseim and B. Verfürth. Math. Comp., 89(326):2649–2674, 2020. BibTeX DOI
  17. Heterogeneous multiscale method for the Maxwell equations with high contrast. B. Verfürth. ESAIM Math. Model. Numer. Anal., 53(1):35–61, 2019. BibTeX DOI
  18. Numerical homogenization of H(curl){\bf {H}}(\rm curl)-problems. D. Gallistl, P. Henning, and B. Verfürth. SIAM J. Numer. Anal., 56(3):1570–1596, 2018. BibTeX DOI
  19. A new heterogeneous multiscale method for the Helmholtz equation with high contrast. M. Ohlberger and B. Verfürth. Multiscale Model. Simul., 16(1):385–411, 2018. BibTeX DOI
  20. Localized Orthogonal Decomposition for two-scale Helmholtz-type problems. M. Ohlberger and B. Verfürth. AIMS Mathematics, 2(3):458–478, 2017. BibTeX DOI
  21. A new heterogeneous multiscale method for time-harmonic Maxwell's equations. P. Henning, M. Ohlberger, and B. Verfürth. SIAM J. Numer. Anal., 54(6):3493–3522, 2016. BibTeX DOI

Conference Proceedings

  1. From domain decomposition to homogenization theory. D. Peterseim, D. Varga, and B. Verfürth. In Domain decomposition methods in science and engineering XXV, volume 138 of Lect. Notes Comput. Sci. Eng., pages 29–40. Springer, Cham, 2020. BibTeX DOI
  2. Computational multiscale method for nonlinear monotone elliptic equations. B. Verfürth. In Oberwolfach Reports, number 35. 2019. BibTeX PDF
  3. Numerical homogenization for indefinite H(curl)-problems. B. Verfürth. In Proceedings of Equadiff 2017 conference, 137–146. 2017. BibTeX URL
  4. Analysis of multiscale methods for time-harmonic maxwell's equations. P. Henning, M. Ohlberger, and B. Verfürth. In Proc. Appl. Math. Mech., volume 16, 559–560. 2016. BibTeX DOI

Theses

  1. Numerical multiscale methods for Maxwell's equations in heterogeneous media. B. Verfürth. PhD thesis, WWU Münster, 2018. BibTeX PDF