Numerical Methods III (FSI-SN3)

Academic year 2023/2024
Supervisor: doc. Ing. Petr Tomášek, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:

The aim of the course is to acquaint students with the mathematical principles of the finite element method and an understanding of algorithmization and standard programming techniques used in its implementation.
Learning outcomes and competences:

In the course Numerical Methods III, students will be made familiar with the finite element method and its mathematical foundations and use this knowledge in several individual projects.
Prerequisites:

Differential and integral calculus for multivariable functions. Fundamentals of functional analysis. Partial differential equations. Numerical methods, especially interpolation, quadrature and solution of systems of ODE. Programming in MATLAB.
Course contents:

The course gives an introduction to the finite element method as a general computational method for solving differential equations approximately. Throughout the course we discuss both the mathematical foundations of the finite element method and the implementation of the involved algorithms.

The focus is on underlying mathematical principles, such as variational formulations of differential equations, Galerkin finite element method and its error analysis. Various types of finite elements are introduced.
Teaching methods and criteria:

The course is taught in the form of lectures explaining the basic principles and theory of the discipline. Seminars are focused on practical topics presented in the lectures.
Assesment methods and criteria linked to learning outcomes:

Graded course-unit credit is awarded on the following conditions: Active participation in practicals and elaboration of assignments. Participation in the lessons may be reflected in the final mark.

If we measure the classification success in percentage points, then the grades are: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.
Controlled participation in lessons:

Attendance at lectures is recommended, attendance at seminars is required. Absence from lessons may be compensated by the agreement with the teacher supervising the seminars.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Computer-assisted exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture

The Finite Element Method in 1D:



  • Variational Formulation

  • Finite Element Approximation

  • Derivation of a Linear System of Equations

  • Computer Implementation

  • A Priori Error Estimate

  • A Posteriori Error Estimate & Adaptive Finite Element Methods


The Finite Element Method in 2D:



  • Variational Formulation

  • Finite Element Approximation

  • Derivation of a Linear System of Equations

  • The Isoparametric Mapping

  • Different Types of Finite Elements

  • Computer Implementation (Data Structuring, Mesh Generation)


Time-Dependant Problems


Abstract Finite Element Analysis



  • Functional Spaces

  • Abstract Variational Problem & Galerkin Method

  • The Lax-Milgram Lemma

  • Galerkin Orthogonality, Best Approximation Property
    Computer-assisted exercise

Seminars will follow the lectures. Students work on assigned projects under the guidance of an instructor.
Literature - fundamental:
1. S.C. Brenner, L.R. Scott: The Mathematical Theory of Finite Element Methods, Springer-Verlag, 2002.
2. L. Čermák: Algoritmy metody konečných prvků, [on-line], available from: http://mathonline.fme.vutbr.cz/Numericke-metody-III/sc-1151-sr-1-a-142/default.aspx.
3. K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations, Cambridge University Press, 1996.
4. A. Ern, J.-L. Guermond: Theory and Practice of Finite Elements, Springer Series in Applied Mathematical Sciences, Vol. 159 (2004) 530 p., Springer-Verlag, New York
4. C. Jonson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1995.
5. P. Knabner, L. Angermann: Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, 2003.
6. M. G. Larson, F. Bengzon: The Finite Element Method: Theory, Implementation, and Applications, Springer, 2013.
Literature - recommended:
8. A. Ženíšek: Matematické základy metody konečných prvků, [on-line], available from: http://mathonline.fme.vutbr.cz/Numericke-metody-III/sc-1151-sr-1-a-142/default.aspx.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-MAI-P full-time study --- no specialisation -- GCr 4 Compulsory 2 1 W