# Associated Project 5 • Numerical analysis of multiscale methods

The propagation of electromagnetic or acoustic waves in heterogeneous media may lead to unusual and astonishing effects such as negative refraction, flat lenses, etc. Typically, the involved materials exhibit a multiscale structure which cannot be resolved by direct numerical simulations even on today's computational architectures. Yet, numerical simulations and mathematical understanding are indispensible tools for the study of the described phenomena due to the complexity of actual experiments. Numerical multiscale methods deliver a coarse-scale (i.e., over the whole material bulk) representation of the solution by an appropriate local upscaling of the fine-scale material features. This is achieved by a decomposition of the solution into a macroscopic part, which can be discretized on a coarse mesh, and a fine-scale contribution. The overall goal of this project is the design and numerical analysis of computational multiscale methods for partial differential equations (PDEs) with general unstructured coefficients.

In the past, we have developed multiscale methods for time-harmonic wave propagation problems [GHV18], in particular with a focus on media with a high-contrast [OSUV20, PV20, Ver19, OV18]. In this project, we focus on Numerical homogenization for nonlinear problems and the numerical treatment of Multiscale coefficients with an additional parameter dependency. Besides these major research foci, we developed and analyzed numerical multiscale methods for sign-changing coefficients [CFV21], which occur in the context of metamaterials, and for PDEs with multiscale dynamic boundary conditions [AV21], which model thin heterogeneous layers.

### Numerical homogenization for nonlinear problems

In the case of high intensities, such as in laser applications, linear material laws are no longer accurate enough, but nonlinear models are employed. The combination of nonlinearities and multiscale features poses a huge computational challenge as many computational multiscale methods rely on linear arguments. We derived and analyzed two new approaches for the linearized and localized construction of problem-adapted basis functions.

The first approach constructs the multiscale basis by a priori linearization. That means that we choose and fix a linearization of the nonlinearity and compute a multiscale basis by solving several small and local linear problems. Collecting the computed basis functions, their span yields a (low-dimensional) multiscale space. This space is then used in a Galerkin method for the nonlinear problem. To solve the resulting nonlinear problem, which is of small dimension, an iteration scheme is applied. This approach is described and analyzed for strictly monotone problems in [Ver22a].

The second approach first chooses an iteration scheme to solve the nonlinear problem. In each iteration step, a linear problem has to be solved. For the latter, the design of an appropriate multiscale space is well studied. Altogether, this results in an iterative multiscale method with a new discrete space in each iteration. To reduce the computational effort, we employ an adaptive strategy which selects the basis functions to be newly computed in each step. This approach is studied for the nonlinear Helmholtz equation in [MV22b].

Besides the multiscale context, we also studied higher-order finite element methods for the nonlinear Helmholtz equation in [Ver22b].

### Multiscale coefficients with an additional parameter dependency

The multiscale methods considered in this project rely on problem-adapted basis functions. This implies that the multiscale basis has to be re-computed every time the coefficient of the PDE changes. However, the main computational effort lies in the calculation of these basis functions, so that this re-computation is cumbersome for problems with many similar coefficients. We have designed new strategies for two cases where the multiscale coefficient depends on an additional parameter, namely randomly perturbed and time-dependent coefficients.

Imperfections and defects in materials due to mistakes in their fabrication process can be modeled by randomly perturbed PDE coefficients. To assess the impact of such defects on the material properties, the PDE needs to be solved for many different realizations of possible defects. We developed an offline-online strategy to compute coarse-scale solutions in this situation. In the offline phase, which is exerted only once, we pre-calculate local stiffness matrix for a clever selection of defects. In the online phase, which is run for each realization in a Monte Carlo setting, we combine the stored offline matrices in a cheap and fast way to assemble a global stiffness matrix. We showed good approximation for moderate defect probabilities and numerically observed computational advantages of the approach for small sample sizes, see [MV22a].

Time-dependent multiscale coefficients play a role in the upcoming field of time-modulated metamaterials. In our numerical multiscale method, the problem-adapted spaces become time-dependent in this situation. We proved a priori error estimates for fully discrete (i.e., discretized in space and time) multiscale methods. Again, we would need to compute new multiscale basis functions in each time step. Therefore, we employed an adaptive strategy to reuse part of the previous multiscale basis. This allows to efficiently tackle wave equations with slow temporal dynamics in the coefficients in [MV22c].

## Publications

1. and . Numerical upscaling for wave equations with time-dependent multiscale coefficients. Multiscale Model. Simul., 20(4):1169–1190, October . [preprint] [files] [bibtex]

2. and . Multiscale scattering in nonlinear Kerr-type media. Math. Comp., 91(336):1655–1685, July . [preprint] [bibtex]

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

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

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

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

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

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

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

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

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

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Dr. Postdoctoral and doctoral researcher