Associated Project 5 • Numerical analysis of multiscale methods

The overall goal of this project is the design and numerical analysis of computational multiscale methods for partial differential equations with general rough and unstructured coefficients and potentially also nonlinearities. Materials with multiscale structures appear in many applications and pose a huge challenge for numerical simulations because fine material features cannot be computationally resolved even with modern computer resources. Numerical multiscale methods rely on the decomposition of the exact solution into a macroscopic and a fine-scale contribution. Hence, the macroscopic or global behavior of the solution can be faithfully approximated with a coarse mesh. One special focus of this project is wave propagation in heterogeneous media, which can lead to unusual and astonishing effects such as negative refraction, flat lenses, etc.

Publications

1. B. Maier and B. Verfürth. Numerical upscaling for wave equations with time-dependent multiscale coefficients. Multiscale Model. Simul., 20(4):1169–1190, October 2022. URL https://doi.org/10.1137/21M1438244. [preprint] [files] [bibtex]

2. R. Maier and B. Verfürth. Multiscale scattering in nonlinear Kerr-type media. Math. Comp., 91(336):1655–1685, July 2022. URL https://doi.org/10.1090/mcom/3722. [preprint] [bibtex]

3. B. Verfürth. Numerical homogenization for nonlinear strongly monotone problems. IMA J. Numer. Anal., 42(2):1313–1338, April 2022. URL https://doi.org/10.1093/imanum/drab004. [preprint] [files] [bibtex]

4. A. Målqvist and B. Verfürth. An offline-online strategy for multiscale problems with random defects. ESAIM: Math. Model. Numer. Anal., 56(1):237–260, March 2022. URL https://doi.org/10.1051/m2an/2022006. [preprint] [files] [bibtex]

5. T. Chaumont-Frelet and B. Verfürth. A generalized finite element method for problems with sign-changing coefficients. ESAIM: Math. Model. Numer. Anal., 55(3):939–967, May 2021. URL https://doi.org/10.1051/m2an/2021007. [preprint] [bibtex]

6. R. Altmann and B. Verfürth. A multiscale method for heterogeneous bulk-surface coupling. Multiscale Model. Simul., 19(1):374–400, February 2021. URL https://doi.org/10.1137/20M1338290. [preprint] [files] [bibtex]

7. M. Ohlberger, B. Schweizer, M. Urban, and B. Verfürth. Mathematical analysis of transmission properties of electromagnetic meta-materials. Netw. Heterog. Media, 15(1):29–56, March 2020. URL https://doi.org/10.3934/nhm.2020002. [preprint] [bibtex]

8. D. Peterseim and B. Verfürth. Computational high frequency scattering from high-contrast heterogeneous media. Math. Comp., 89(326):2649–2674, March 2020. URL https://doi.org/10.1090/mcom/3529. [preprint] [bibtex]

9. B. Verfürth. Heterogeneous multiscale method for the Maxwell equations with high contrast. ESAIM: Math. Model. Numer. Anal., 53(1):35–61, March 2019. URL https://doi.org/10.1051/m2an/2018064. [preprint] [bibtex]

10. D. Gallistl, P. Henning, and B. Verfürth. Numerical homogenization of ${\bf{H}}(\rm curl)$-problems. SIAM J. Numer. Anal., 56(3):1570–1596, June 2018. URL https://doi.org/10.1137/17M1133932. [preprint] [bibtex]

11. M. Ohlberger and B. Verfürth. A new heterogeneous multiscale method for the Helmholtz equation with high contrast. Multiscale Model. Simul., 16(1):385–411, March 2018. URL https://doi.org/10.1137/16M1108820. [preprint] [bibtex]

Preprints

12. B. Verfürth. Higher-order finite element methods for the nonlinear Helmholtz equation. CRC 1173 Preprint 2022/47, Karlsruhe Institute of Technology, September 2022. [files] [bibtex]