401-3651-00L  Numerical Analysis for Elliptic and Parabolic Partial Differential Equations

SemesterAutumn Semester 2020
LecturersC. Schwab
Periodicityyearly recurring course
Language of instructionEnglish
CommentCourse audience at ETH:
3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students.
Other ETH-students are advised to attend the course
"Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester.



Courses

NumberTitleHoursLecturers
401-3651-00 VNumerical Analysis for Elliptic and Parabolic Partial Differential Equations4 hrs
Tue10:15-12:00HG D 5.2 »
Wed12:15-14:00HG D 5.2 »
C. Schwab
401-3651-00 UNumerical Analysis for Elliptic and Parabolic Partial Differential Equations1 hrs
Wed09:15-10:00ML F 40 »
C. Schwab

Catalogue data

AbstractThis course gives a comprehensive introduction into the numerical treatment of linear and nonlinear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods.
ObjectiveParticipants of the course should become familiar with
* concepts underlying the discretization of elliptic and parabolic boundary value problems
* analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems
* methods for the efficient solution of discrete boundary value problems
* implementational aspects of the finite element method
ContentThe course will address the mathematical analysis of numerical solution methods
for linear and nonlinear elliptic and parabolic partial differential equations.
Functional analytic and algebraic (De Rham complex) tools will be provided.
Primal, mixed and nonstandard (discontinuous Galerkin, Virtual, Trefftz) discretizations will be analyzed.

Particular attention will be placed on developing mathematical foundations
(Regularity, Approximation theory) for a-priori convergence rate analysis.
A-posteriori error analysis and mathematical proofs of adaptivity and optimality
will be covered.
Implementations for model problems in MATLAB and python will illustrate the
theory.

A selection of the following topics will be covered:

* Elliptic boundary value problems
* Galerkin discretization of linear variational problems
* The primal finite element method
* Mixed finite element methods
* Discontinuous Galerkin Methods
* Boundary element methods
* Spectral methods
* Adaptive finite element schemes
* Singularly perturbed problems
* Sparse grids
* Galerkin discretization of elliptic eigenproblems
* Non-linear elliptic boundary value problems
* Discretization of parabolic initial boundary value problems
LiteratureSUPPLEMENTARY Literature (core material will be in lecture notes)


Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. xviii+397 pp.

A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods,
Springer Applied Mathematical Sciences Vol. 159, Springer,
1st Ed. 2004, 2nd Ed. 2015.

R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013

Additional Literature:
D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007).
(Also available in German.)

Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp.

D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications,
Springer, 2012 [DOI: 10.1007/978-3-642-22980-0]

V. Thomee: Galerkin Finite Element Methods for Parabolic Problems,
SECOND Ed., Springer Verlag (2006).
Prerequisites / NoticePractical exercises based on MATLAB

Former title of the course unit: Numerical Methods for Elliptic and Parabolic Partial Differential Equations

Performance assessment

Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course
ECTS credits10 credits
ExaminersC. Schwab
Typesession examination
Language of examinationEnglish
RepetitionThe performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Mode of examinationoral 30 minutes
This information can be updated until the beginning of the semester; information on the examination timetable is binding.

Learning materials

 
Main linkInformation
Only public learning materials are listed.

Groups

401-3651-00 UNumerical Analysis for Elliptic and Parabolic Partial Differential Equations
GroupG-01
Wed09:15-10:00ML F 40 »

Restrictions

There are no additional restrictions for the registration.

Offered in

ProgrammeSectionType
Doctoral Department of MathematicsGraduate SchoolWInformation
Mathematics MasterCore Courses: Applied Mathematics and Further Appl.-Oriented FieldsWInformation