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Lecturer(s)
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Hozman Jiří, RNDr. Mgr. Ph.D.
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Course content
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Lectures: 1. Introduction to theory of Sobolev spaces. Fundamental ideas ideas of the finite element method (FEM). Definition of weak solution of the Poisson's equation with Dirichlet boundary conditions. 2. Abstract elliptic variational problem. Classical and weak solution, Lax-Milgram lemma. Finite element spaces, basis. 3. Triangulation of computational domain, construction of finite element space. Examples of finite elements defined on simplices and rectangles. Barycentric coordinates. 4. General definition of the finite element. Affine equivalence of finite elements, 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. 8. Error estimates in L2 norm. Non-homogeneous boundary conditions. 9. Error estimates of approximate solution. Influence of numerical integration and approximation of boundary of computational domain on error of approximate solution. 10. Resulting systems of linear algebraic equations corresponding to the discrete elliptic problems and their properties. Basic method for solving the linear algebraic systems. 11. Finite element discretization of parabolic problems. Linear parabolic equation. Semi-discretization by virtue of conforming finite element method, analysis of system of ordinary differential equations (ODE) arising from semidiscretization. 12. Numerical methods for ODE and their analysis - stability, consistency, convergency, order of mehod. 13. Finite element discretization of hyperbolic problems of the second order. 14. Reserve. Practice: The material explained at the previous week lecture is practised.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 56 hours per semester
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Learning outcomes
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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.
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Prerequisites
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Fundamentals of numerical mathematics.
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Assessment methods and criteria
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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.
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Recommended literature
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Brenner S. - Scott R.:. The mathematical theory of finite element methods. 1994.
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Ciarlet, P.G.:. The finite element method for elliptic problems. 1978.
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Haslinger, J.:. Metoda konečných prvků pro řešení eliptických rovnic a nerovnic.. Praha, SPN, 1980.
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Haslinger, J.:. Řešení variačních rovnic a nerovnic, skriptum.. MF UK, Praha, 1983.
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Rektorys, K.:. Variační metody v inženýrských problémech a v problémech matematické fyziky.. Praha, 1974.
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Šolín, P. - Segeth, K. - Doležel, I.:. Higher-Order Finite Element Methods.. Boca Raton, FL, Chapman & Hall/CRC, 2004.
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Šolín, P.:. Partial Differential Equations and the Finite Element Method.. Hoboken, NJ, Wiley, 2005.
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