# 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. . Numerical homogenization for nonlinear strongly monotone problems. IMA J. Numer. Anal., 42(2):1313–1338, April . [preprint] [files] [bibtex]

2. and . An offline-online strategy for multiscale problems with random defects. ESAIM Math. Model. Numer. Anal., 56(1):237–260, March . [preprint] [files] [bibtex]

3. and . Multiscale scattering in nonlinear Kerr-type media. Math. Comp., 1–31, February . Online first. [preprint] [bibtex]

4. and . A generalized finite element method for problems with sign-changing coefficients. ESAIM: M2AN, 55(3):939–967, May . [preprint] [bibtex]

5. and . A multiscale method for heterogeneous bulk-surface coupling. Multiscale Model. Simul., 19(1):374–400, February . [preprint] [files] [bibtex]

6. , , , and . Mathematical analysis of transmission properties of electromagnetic meta-materials. Netw. Heterog. Media, 15(1):29–56, March . [preprint] [bibtex]

7. and . Computational high frequency scattering from high-contrast heterogeneous media. Math. Comp., 89(326):2649–2674, March . [preprint] [bibtex]

8. . Heterogeneous multiscale method for the Maxwell equations with high contrast. ESAIM Math. Model. Numer. Anal., 53(1):35–61, March . [preprint] [bibtex]

9. , , and . Numerical homogenization of ${\bf{H}}(\rm curl)$-problems. SIAM J. Numer. Anal., 56(3):1570–1596, June . [preprint] [bibtex]

10. and . A new heterogeneous multiscale method for the Helmholtz equation with high contrast. Multiscale Model. Simul., 16(1):385–411, March . [preprint] [bibtex]

## Preprints

1. and . Numerical upscaling for wave equations with time-dependent multiscale coefficients. CRC 1173 Preprint 2021/34, Karlsruhe Institute of Technology, July . [files] [bibtex]

Former staff
Name Title Function
Dr. Postdoctoral and doctoral researcher