Associated Project 5 • Numerical analysis of multiscale methods

Illustration
Wave in a multiscale medium.

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. and . A generalized finite element method for problems with sign-changing coefficients. ESAIM: M2AN, 55(3):939–967, May . URL https://doi.org/10.1051/m2an/2021007. [preprint] [bibtex]

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

  3. . Numerical homogenization for nonlinear strongly monotone problems. IMA J. Numer. Anal., 1–26, February . URL https://doi.org/10.1093/imanum/drab004. Online first. [preprint] [files] [bibtex]

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

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

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

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

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

Preprints

  1. and . An offline-online strategy for multiscale problems with random defects. CRC 1173 Preprint 2021/7, Karlsruhe Institute of Technology, February . [files] [bibtex]

  2. and . Multiscale scattering in nonlinear Kerr-type media. CRC 1173 Preprint 2020/36, Karlsruhe Institute of Technology, November . [bibtex]