New homepage
Since October 1, 2018, I am working as research associate (PostDoc) at the University of Augsburg. Please visit my homepage there for up-to-date information.
Research Foci
- Multiscale Finite Element Methods
- (Numerical) Homogenization
- Time-harmonic Maxwell's equations
Doctoral AbstractThesis
Numerical multiscale methods for Maxwell's equations in heterogeneous media
- Supervisor
- Professor Dr. Mario Ohlberger
- Doctoral Subject
- Mathematik
- Doctoral Degree
- Dr. rer. nat.
- Awarded by
- Department 10 – Mathematics and Computer Science
In this thesis we introduce new numerical multiscale methods for problems arising from time-harmonic Maxwell's equations in heterogeneous media. These problems are used to model electromagnetic wave propagation, for instance, in the context of photonic crystals. Such materials can exhibit unusual optical properties, where we are in particular interested in negative refraction. Although this phenomenon and its effects have been studied in a lot of physical experiments, the mathematical understanding of this topic is still in its infancy.As a first step, we consider elliptic problems with heterogeneous (rapidly varying) coefficients involving the double application of the curl as differential operator. The corresponding solutions typically admit very low regularity and conventional numerical schemes have arbitrarily bad convergence rates. For locally periodic problems, we suggest a Heterogeneous Multiscale Method and prove a priori error estimates.Numerical experiments are given to confirm the convergence rates and to validate the applicability of the method. In order to cope with more general coefficients, we construct a generalized finite element method in the spirit of the Localized Orthogonal Decomposition. The method decomposes the exact solution into a coarse-scale part (spanned by standard finite element functions) and a fine-scale part. A stable corrector operator, which is quasi-local and thus can be computed efficiently, allows to represent and extract necessary fine-scale features of the solution. We show that this construction enjoys optimal approximation properties in energy and dual norms.As the next and even more challenging step towards negative refraction, we consider (indefinite) scattering problems with periodic high contrast coefficients. Here, periodically distributed inclusions are associated with a much smaller material coefficient (scaled like the square of the periodicity length) than the rest of the scatterer. Homogenization results show that the high contrast leads to unusual effective parameters in the homogenized equation. Consequently, wave propagation inside the scatterer is physically forbidden for certain wavenumbers; this effect is called a band gap. In the analysis of the homogenized formulation, we particularly prove new wavenumber-explicit stability estimates for solutions to the Helmholtz and Maxwell equations. As numerical discretization scheme we propose a Heterogeneous Multiscale Method, for which we show inf-sup stability, quasi-optimality, and a priori error estimates. These results are obtained under a (standard) resolution condition between the wavenumber and the mesh size. Numerical experiments confirm the convergence rates and give an explanation of the physical phenomenon of band gaps.
The whole PhD thesis is available here.Honors
- Dissertation Prize of the Faculty of Mathematics and Computer Science – University of Münster
Teaching
- Seminar: Master-und Oberseminar zu effizienten numerischen Methoden [104435]
(in cooperation with Dr. Stephan Rave, , Dr. Felix Schindler and ) - Seminar: Master-und Oberseminar zu effizienten numerischen Methoden [102385]
(in cooperation with Dr. Stephan Rave, Dr. Felix Schindler, , Prof. Dr. Mario Ohlberger and ) - Practice: Übungen zur Numerischen Analysis [102373]
(in cooperation with Tobias Leibner and Prof. Dr. Mario Ohlberger)
- Seminar: Master-und Oberseminar zu effizienten numerischen Methoden [104435]
Project
- Wave propagation in periodic structures and negative refraction mechanisms ()
Individual Granted Project: | DFG - Individual Grants Programme | Project Number: OH 98/6-1
- Wave propagation in periodic structures and negative refraction mechanisms ()
Publications
- Ohlberger, Mario Schweizer Ben, and Urban, Maik Verfürth Barbara. . “Mathematical analysis of transmission properties of electromagnetic meta-materials.” Networks and Heterogeneous Media 15 (1): 29–56. doi: 10.3934/nhm.2020002.
- Verfürth, B. in Kürze. “Heterogeneous Multiscale Method for the Maxwell equations with high contrast.” ESAIM Math. Model. Numer. Anal. xx.
- Gallistl, D, Henning, P, and Verfürth, B. . “Numerical homogenization of H(curl)-problems.” SIAM J. Numer. Anal. 56 (3): 1570–1596. doi: 10.1137/17M1133932.
- Ohlberger, M, and Verfürth, B. . “A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast.” Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal 16 (1): 385–411. doi: 10.1137/16M1108820.
- Verfürth, B. . “Numerical multiscale methods for Maxwell's equations in heterogeneous media.” Dissertation thesis, Universität Münster.
- Ohlberger, M, and Verfürth, B. . “Localized Orthogonal Decomposition for two-scale Helmholtz-type problems.” AIMS Mathematics 2 (3): 458–478. doi: 10.3934/Math.2017.2.458.
- Verfürth, B. . “Numerical homogenization for indefinite H(curl)-problems.” in Proceedings of Equadiff 2017 Conference, edited by K Mikula, D Sevcovic and J Urban.
- Verfürth, B. . “Heterogeneous Multiscale Method for a Helmholtz problem with high contrast.” contribution to the Winter school on Numerical Analysis of Multiscale Problems, HIM Bonn, Germany
- Henning, P, Ohlberger, M, and Verfürth, B. . “Analysis of multiscale methods for time-harmonic Maxwell's equations.” Proc. Appl. Math. Mech. 16 (1): 559–560. doi: 10.1002/pamm.201610268.
- Henning, P, Ohlberger, M, and Verfürth, B. . “A new Heterogeneous Multiscale Method for time-harmonic Maxwell's equations.” SIAM J. Numer. Anal. 54 (6): 3493–3522. doi: 10.1137/15M1039225.
Scientific Talks
- Verfürth, Barbara : “Numerical multiscale methods for Maxwell's equations in complex media”. Interplay of multiscale data assimilation and data science with advanced PDE discretizations, Erwin-Schrödinger-Institut Wien, Österreich, .
- Verfürth, Barbara : “Maxwell’s equations in periodic microstructures with high contrast”. European Finite Element Fair, Heidelberg, Deutschland, .
- Verfürth, Barbara : “Numerical multiscale methods for Maxwell's equations in periodic media”. 4th Applied Math Symposium Münster, Münster, Deutschland, .
- Verfürth, Barbara : “Numerical multiscale methods for Maxwell's equations”. Joint Annual Meeting of GDM and DMV, Paderborn, Deutschland, .
- Verfürth, Barbara : “Multiscale method for waves in periodic structures with high contrast”. Oberseminar, Institut für Mathematik, Universität Augsburg, .
- Verfürth, Barbara : “Multiscale methods for waves in periodic structures”. ENUMATH 2017, Voss, Norwegen, .
- Verfürth, Barbara : “Numerical homogenization for electromagnetic wave propagation”. Equadiff 2017, Bratislava, Slowakei, .
- Verfürth, Barbara : “Multiscale Methods for time-harmonic electromagnetic waves”. iRTG Seminar, SFB 1173 Wellenphänomene, Karlsruhe Institut für Technologie (KIT), Deutschland, .
- Verfürth, Barbara : “Heterogeneous Multiscale Methods for time-harmonic waves in periodic media”. Junior Seminar, Hausdorff-Trimesterprogramm "Multiscale Problems", HIM Bonn, Deutschland, .
- Verfürth, Barbara : “Analysis of a multiscale method for highly heterogeneous Helmholtz problems”. Workshop METAMATH Waves in periodic media and metamaterials, Cargèse, Frankreich, .
- Verfürth, Barbara : “Multiscale methods for highly heterogeneous Helmholtz problems”. European Finite Element Fair, Bonn, Deutschland, .
- Verfürth, Barbara : “Analysis of multiscale methods for time-harmonic Maxwell's equations”. Joint Annual Meeting of GAMM and DMV, Braunschweig, Deutschland, .