Please note! Course description is confirmed for two academic years, which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.
LEARNING OUTCOMES
Credits: 5
Schedule: 19.04.2021 - 04.06.2021
Teacher in charge (valid 01.08.2020-31.07.2022): Heikki Apiola
Teacher in charge (applies in this implementation): Rolf Stenberg
Contact information for the course (valid 15.03.2021-21.12.2112):
Lecturer: Rolf Stenberg. rolf.stenberg@aalto.fi. 050-5935984
CEFR level (applies in this implementation):
Language of instruction and studies (valid 01.08.2020-31.07.2022):
Teaching language: Finnish
Languages of study attainment: Finnish, Swedish
CONTENT, ASSESSMENT AND WORKLOAD
Content
Applies in this implementation:
The basic finite element method is for problems posed as the minimisation of a quadratic functional.
The FE method leads to a discrete piecewise polynomial solution which is the best approximation in the energy norm given by the functional.
There are, however, numerous problems for which the minimisation is done subject to a linear constraint. This leads to interesting and difficult problems, both regarding the governing system of partial differential equations and their approximation.
For the partial differential differential equations the challenge is to find the right functional analytic framework for which the existence, uniqueness and stability are valid. It turns out that the finite element method should be carefully designed in order that the conditions of existence, uniqueness and stability are inherited in the discretisation.
In this course the theory for this class of problems is studied. We start by introducing the so-called Clement interpolation operator which is a central tool for the theory. The first application of it is in a posteriori error analysis which leads to adaptive finite element methods, i.e. methods with mesh refinement for optimising the accuracy. Then we will study the Stokes equations which is the most well-known system of this type. The mathematical theory for the governing partial differential equations is developed in detail. Then we study how a stable and optimally convergent FE methods should be designed and mathematically analysed.
The theory is supported by computer exercises.
Assessment Methods and Criteria
Applies in this implementation:
The course is passed by handing in home exercises.
DETAILS
Study Material
Applies in this implementation:
The lecture notes from last year is posted on the course website.
They will be updated during the course.
- Opettaja: Stenberg Rolf