Numerik IV: Elliptic partial differential equations with highly oscillating or random coefficients

News and Important Remarks


Lecture Thu, 10 - 12 WebEx Prof. Dr. Ralf Kornhuber
Tutorial Wed, 10 - 12 WebEX Prof. Dr. Ralf Kornhuber
Exam February 23 2021 WebEX
Second exam April 7, 8 2021 WebEx

General Information


Multiple scales are ubiquitous in a plurality of processes and phenomena and therefore attract considerable attention both in mathematical and natural science research (see, e.g., here). In this lecture we will concentrate on elliptic problems with an highly oscillatory or random behavior due to corresponding coefficient functions. In particular, we will present recent 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 at FU Berlin or in Phase I of the Berlin Mathematical School (BMS).


Basic knowledge on theory and numerical analysis of elliptic pdes as presented, e.g., in the lecture "Numerics of partial differential equations" at FU Berlin.


All participants have to register at KVV in order to have access to course information such as login credentials for the lectures and tutorials. In addition, FU students have to register in the Campus Management in order to receive a certificate in the case of success. Possible external students should contact Ralf Kornhuber for exceptions.

Exercises and Criteria for a Certificate


Unpolished online notes of the lectures can be found here:

Tutorial & Exercises

Oral Exam

Criteria for the Certificate (Übungsschein)

Necessary and sufficient for a certificate are:

The final grade of the certificate is the best of the grades of the exam and of the second exam.



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

  2. Allaire G. (1989). Homogenization of the Stokes flow in a connected porous medium. Asymptotic Analysis 2, 203-222.

  3. Clément, P. (1975). Approximation by finite element functions using local regularization. Revue française d'automatique, informatique, recherche opérationnelle. Analyse numérique, 9(R2), 77-84.

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

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

  6. 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

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

  8. Verfürth, R. (1999). Error estimates for some quasi-interpolation operators. ESAIM: Mathematical Modelling and Numerical Analysis, 33(4), 695-713.


Prof. Dr. Ralf Kornhuber Arnimallee 6, Room 130