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

News and Important Remarks

• Some inconsistencies in the damping factors and optimal convergence rates in lectures 5 and 6 have been corrected.

• We are planning to hold this lecture online. The first lecture in Numerics IV will be held on November 5th 2020 and will take place via Cisco WebEx. Detailed instructions can be found here. You will receive an e-mail with login credentials to lectures and tutorial. Please provide the usual technical requirements for online communication such as headset, camera, bandwidth etc.

Dates

General Information

Description

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).

Prerequisites:

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.

Registration

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 ralf.kornhuber@fu-berlin.de for exceptions.

Exercises and Criteria for a Certificate

Lectures

Unpolished online notes of the lectures can be found here:

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture 4
• Lecture 5
• Lecture 6
• Lecture 7
• Lecture 8
• Lecture 9
• Lecture 10
• Lecture 11
• Lecture 12
• Lecture 13

Tutorial & Exercises

• The tutorial starts November 18. It offers the possibility to discuss and better understand the presented material and exercises. The tutorial is mandatory for active participation.
• Work sheets with exercises will be offered electronically on this web page (see below).
• The exercises involve theoretical and numerical problems.
  that should be solved by teams of at least two and at most three students. Both types of exercises are rated separately.
• Numerical problems should be solved using Matlab or Octave (if Matlab is not available).
• Solutions of both numerical and theoretical problems will be discussed in the tutorial.

Oral Exam

• There will be an an oral exam via Cisco WebEx at the end of the semester with the the opportunity of a second oral exam via Cisco WebEx in April (see Dates).

Criteria for the Certificate (Übungsschein)

Necessary and sufficient for a certificate are:

• individual examination: passing the exam or the second exam
• active participation: solution of sufficiently many exercises (at least 60 % of theoretical and 60 % of programming points) and presentation of at least one solution in the tutorial,
• regular participation: (virtual or physical) presence in lectures and tutorials (not monitored)

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

Exercises

• Exercise 1
• Exercise 2
• Exercise 3
• Exercise 4
• Exercise 5
• Exercise 6

Literature

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.

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