Content:

Multiple scales are ubiquitous in a plurality of processes and phenomena and therefore attract considerable attention both in mathematical and natural science researach (see, e.g., CRC 1114 at FU Berlin). In this lecture we will concentrate on elliptic problems with an highly oscillatory or random behaviour due to corresponding coefficient functions. In particular, we will present new results on a class of variational multiscale methods based on subspace decomposition and adaptive multilevel Monte Carlo (MLMC) methods. If time permits, parabolic problems and random pdes will be also addressed.

Target audience:
Students in the Master Course Mathematics or BMS (Phase I)

Prerequisites:
Basic knowledge on theory and numerics of elliptic pdes as considered, e.g. in the lecture "Numerik von partiellen Differentialgleichungen" (Numerik III) at FU.

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Literatur

[1]  Allaire, G. (1992). Homogenization and two-scale convergence. SIAM Journal on Mathematical Analysis, 23(6), 1482-1518.

[2] Målqvist, A., & Peterseim, D. (2014). Localization of elliptic multiscale problems. Mathematics of Computation, 83(290), 2583-2603

[3] Kornhuber, R., & Yserentant, H. (2016). Numerical homogenization of elliptic multiscale problems by subspace decomposition. Multiscale Modeling & Simulation, 14(3), 1017-1036.

[4] Kornhuber, R., Peterseim, D., & Yserentant, H. (2018). An analysis of a class of variational multiscale methods based on subspace decomposition. Mathematics of Computation, 87(314), 2765-2774

[5] Le Maître, O., & Knio, O. M. (2010). Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer Science & Business Media.