Description:

The course is devoted to the mathematical theory of the finite element method numerically solving boundary-value and initial-boundary-value problems for partial differential equations. Mathematical properties of the method are explained. The approximation error estimates are derived.

Contents:

1. Weak solution of boundary-value 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. Bramble-Hilbert lemma
8. Global interpolation error
9. Mathematical features of the FEM and details of use
10. Examples of software packages based on FEM

Seminar contents:

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.

Recommended literature:

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, North-Holland, 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 non-smooth 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

Keywords:

Boundary-value problems for partial differential equations, finite-element method, Galerkin method, Bramble-Hilbert lemma, interpolation error.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar