Course detail

Numerical Methods III

FSI-SN3-AAcad. 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

English

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

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
S.C. Brenner, L.R. Scott: The Mathematical Theory of Finite Element Methods, Springer-Verlag, 2002.
P. Knabner, L. Angermann: Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, 2003.
C. Jonson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1995.

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.
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.

Classification of course in study plans

  • Programme N-MAI-A 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 first four lectures will be devoted to the explanation of the algorithm for solution of the model problem of type "stationary heat conduction" in a plane polygonal domain using linear triangular finite elements. This enables students from the very beginning of practicals to start experimenting with code programming. Only the following lectures will concentrate on the mathematical theory of finite elements.
1. Classical and variational formulation, triangulation, piecewise linear functions.
2. Discrete variational formulation, elementary matrices and vectors.
3. Elementary matrices and vectors - continuation.
4. Assembly of global system of equations, its solution, postprocessing.
5. Selected pieces of knowledge of functional analysis. The space W^k_2.
6. Traces of functions from the space W^k_2. Friedrich's and Poincare's inequality.
7. Bramble-Hilbert's lemma. Sobolev's imbedding theorem.
8. Formal equivalence of the elliptic boundary value problem and the related variational problem.
9. Finite element spaces of Lagrange's type. Definition of approximate solution. Existence and uniqueness theorem.
10. Transformation of a general triangle onto the reference triangle. Relations between norms on the general triangle and on the reference triangle.
11. Interpolation theorem.
12. Numerical integration.
13. Adaptivity in FEM.

Computer-assisted exercise

13 hours, compulsory

Teacher / Lecturer

Ing. Petra Rozehnalová, Ph.D.

Syllabus

Practicals will take place in a computer lab with the support of the MATLAB and Visual Studio. The algorithm for the elliptic problem will be explained during the first four lessons. The algorithms for the parabolic, hyperbolic and eigenvalue problems will be explained in brief on practicals. It is supposed that students will work individually with lecture notes (containing detailed descriptions of algorithms). Students are also expected to create and debug individually their own MATLAB programs.
1-2. Programming tools, first introduction.
3-4. Further details, preparation for writing of the program for solution of an elliptic problem (stationary heat conduction).
5-6. Developing the program for an elliptic problem. Explanation of the algorithm for the solution of the parabolic problem (nonstationary heat conduction).
7-8. Developing the program for a parabolic problem. Explanation of the algorithm for the solution of the hyperbolic problem (membrane vibrations).
9-10. Developing the program for a hyperbolic problem. Explanation of the algorithm for the solution of the eigenvalue problem.
11-12. Developing the program for an eigenvalue problem.
13. Teacher's reserve.