Multiscale methods for elliptic problems

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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 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 stochastic 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 taught, e.g. in the lecture "Numerik von partiellen Differentialgleichungen" (Numerik III)

Registration

Depending on your program of study, registration in the Campus Management is mandatory. In addition, all participants should register at the KVV.

Exercises and Criteria for a Certificate

Tutorial & Exercises

• The tutorial offers the possibility to discuss and better understand the presented material and exercises. Furthermore solutions of exercises are presented by the students.
• A sheet with exercises will be handed out each Friday during the lecture. These exercises are also available electronically on this web page (see below).
• The exercises are intended to be solved by teams of two members.
• The exercises consist of theoretical and numerical problems. The latter should be solved using Matlab (available at the students computer pool at the institute). Both types of exercises are rated seperately.
• The solutions have to be finished before the tutorial 11 days after they were handed out.
• The solutions of the numerical problems should be delivered by email to adjurdjevac{at}zedat.fu-berlin.de Note that a complete solution for a numerical experiment consists of a running Matlab code, a program executing the required test runs, and protocols of the execution of these test runs. Delivering a correct and running code without knowing what is going on in the code will be regarded and rated as attempt of deception.
• The solutions of the theoretical problems have to be presented in the tutorial on request (without regard to preference and presence) by a member of a given group.

Exam

• There will be a 20 minutes oral exam on Tuesday, 21.02.2017

Criteria for the Certificate

Neccessary and sufficient for a certificate are:

• passing the oral exam
• active participation: at least as much success as failure in the presentation of the solution of theoretical problems and 50% of the maximal number of points for numerical problems
• constant participation: 85% attendence in the tutorial (not checked)

Certificates are graded according to the result of the exam.

Exercises

Literature

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