|
Lecturer(s)
|
-
Hozman Jiří, RNDr. Mgr. Ph.D.
|
|
Course content
|
Lectures: 1. Introduction to theory of Sobolev spaces. Fundamental ideas ideas of the finite element method (FEM). Definition of the weak solution of the boundary value problem of the second order for various boundary conditions. 2. Abstract elliptic variational problem. Classical and weak solution, Lax-Milgram lemma. Finite element spaces and basis for one-dimensional problems. 3. Triangulation of computational domain, construction of finite element space. Examples of finite elements defined on interval. Barycentric coordinates. 4. General definition of the finite element. Concept of reference finite element and its sense for theoretical considerations and implementation. 5. General definition of the finite element space. Boundary conditions. 6. General considerations on convergence of FEM. Basic error estimates of approximate solutions. 7. Approximate properties of finite element spaces. Convergency of discrete solution of elliptic problems. Practice: 1. Variational formulations of boundary value problems in 1D. Modifications for various boundary conditions. 2. An existence and uniqueness of the weak solution, verification of assumptions of Lax-Milgram lemma. 3. Examples of finite elements defined on interval. 4. Implementation of linear (Lagrange) finite elements in 1D, global representation of a system of linear equations. Modifications for different boundary value problems and various boundary conditions. 5. Implementation of linear (Lagrange) finite elements in 1D, local representation of a system of linear equations. 6. Implementation of quadratic (Lagrange) finite elements in 1D, local representation of a system of linear equations. 7. Reserve.
|
|
Learning activities and teaching methods
|
Monological explanation (lecture, presentation,briefing)
- Class attendance
- 56 hours per semester
|
|
Learning outcomes
|
Mathematical foundations of the finite element method. Variational formulation, discretization, convergence, error estimates and implementation.
Mathematical fundamentals of the finite element method. Application to the solution of boundary value problems for ordinary and partial differential equations. Basic algorithms and survey of software.
|
|
Prerequisites
|
Fundamentals of numerical mathematics.
|
|
Assessment methods and criteria
|
Combined examination
Credit: Active participation on practice. Preparation of an assigned computational semester work on computer. Exam: Written, composed of the theoretical and computational part.
|
|
Recommended literature
|
-
Brenner S. - Scott R.:. The mathematical theory of finite element methods. 1994.
-
Ciarlet, P.G.:. The finite element method for elliptic problems. 1978.
-
Haslinger, J.:. Metoda konečných prvků pro řešení eliptických rovnic a nerovnic.. Praha, SPN, 1980.
-
Haslinger, J.:. Řešení variačních rovnic a nerovnic, skriptum.. MF UK, Praha, 1983.
-
Rektorys, K.:. Variační metody v inženýrských problémech a v problémech matematické fyziky.. Praha, 1974.
-
Šolín, P. - Segeth, K. - Doležel, I.:. Higher-Order Finite Element Methods.. Boca Raton, FL, Chapman & Hall/CRC, 2004.
-
Šolín, P.:. Partial Differential Equations and the Finite Element Method.. Hoboken, NJ, Wiley, 2005.
|