Finite Element Method
Code  Completion  Credits  Range  Language 

01MKP  ZK  3  1P+1C  Czech 
 Garant předmětu:
 Michal Beneš
 Lecturer:
 Michal Beneš
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

The course is devoted to the mathematical theory of the finite element method numerically solving boundaryvalue and initialboundaryvalue problems for partial differential equations. Mathematical properties of the method are explained. The approximation error estimates are derived.
 Requirements:

Basic course of Calculus, Linear Algebra and Numerical Mathematics, variational methods (in the extent of the courses 01MA1, 01MAA24, 01LA1, 01LAA, NM, or 01MA1, 01MAB24, 01LA1, 01LAB2, NMET, VAME held at the FNSPE CTU in Prague).
 Syllabus of lectures:

1. Weak solution of boundaryvalue problem for an elliptic partial differential equation.
2. Galerkin method
3. Basics and features of the FEM
4. Definition and common types of finite elements.
5. Averaged Taylor polynomial
6. Local and global interpolant
7. BrambleHilbert lemma
8. Global interpolation error
9. Mathematical features of the FEM and details of use
10. Examples of software packages based on FEM
 Syllabus of tutorials:

Exercise is merged with the lecture and contains examples of problem formulation, examples on function bases, examples related to the interpolation theory and examples of software packages based on FEM, in particular.
 Study Objective:

Knowledge:
Weak formulation of boundaryvalue and initialboundaryvalue problems for partial differential equations, Galerkin method, basics of FEM, error estimates, applications.
Skills:
Formulation of given problem into the form convenient for FEM, method implementation, application, explanation of results and error assessment.
 Study materials:

Key references:
[1] S. C. Brenner a L. Ridgway Scott, The mathematical theory of finite element methods, New York, Springer 1994
[2] P.G. Ciarlet, The finite element method for elliptic problems, Amsterdam, NorthHolland, 1978
[3] V. Thomée, The Galerkin finite element methods for parabolic problems, LNM 1054, Berlin, Springer, 1984
[4] S. A. Ragab, H. E. Fayed, Introduction to Finite Element Analysis for Engineers, CRC Press, Taylor Francis, 2017
Recommended references:
[5] P. Grisvard, Elliptic problems in nonsmooth domains, Boston, Pitman, 1985
[6] K. Rektorys, Variational methods in engineering and mathematical physics, Praha, Academia 1999 (translated to English)
Media and tools:
Computer training room with OS Windows/Linux and software package FEM
 Note:
 Timetable for winter semester 2023/2024:
 Timetable is not available yet
 Timetable for summer semester 2023/2024:
 Timetable is not available yet
 The course is a part of the following study plans:

 Matematické inženýrství (compulsory course of the specialization)
 Aplikovaná algebra a analýza (elective course)
 Matematické inženýrství (compulsory course in the program)
 Fyzikální elektronika  Počítačová fyzika (PS)