Numerical Methods III (FSI-SN3-A)

Academic year 2023/2024
Supervisor: doc. Ing. Petr Tomášek, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
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 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 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.
Literature - fundamental:
1. 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
2. S.C. Brenner, L.R. Scott: The Mathematical Theory of Finite Element Methods, Springer-Verlag, 2002.
3. P. Knabner, L. Angermann: Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, 2003.
4. C. Jonson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1995.
Literature - recommended:
1. 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.
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.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-MAI-A full-time study --- no specialisation -- GCr 4 Compulsory 2 1 W