Course detail

Numerical Methods III

FSI-SN3Acad. year: 2023/2024

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.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. Ing. Petr Tomášek, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

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.

Rules for evaluation and completion of the course

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.


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.

Aims

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.


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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

S.C. Brenner, L.R. Scott: The Mathematical Theory of Finite Element Methods, Springer-Verlag, 2002. (EN)
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. (CS)
K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential Equations, Cambridge University Press, 1996. (EN)
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 (EN)
C. Jonson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1995. (EN)
P. Knabner, L. Angermann: Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, 2003. (EN)
M. G. Larson, F. Bengzon: The Finite Element Method: Theory, Implementation, and Applications, Springer, 2013. (EN)

Recommended reading

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. (CS)

Classification of course in study plans

  • Programme N-MAI-P Master's, 1. year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

Ing. Petra Rozehnalová, Ph.D.

Syllabus

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

13 hours, compulsory

Teacher / Lecturer

Ing. Petra Rozehnalová, Ph.D.

Syllabus

Seminars will follow the lectures. Students work on assigned projects under the guidance of an instructor.