Course detail

The students should be acquainted with the basics of functional analysis needed to understand the principles of the calculus of variation and non-numeric solutions of initial and boundary problems.

Students will have an overview on advanced methods of mathematical analysis (basic notions of functional analysis, derivatives of a functional, fixed point theorems), methods of calculus of variations and on selected numerical methods for solving of problems for partial differential equations.

Basics of the theory of one- and more-functions. Differentiation and integration of functions.

Functional spaces, the notion of a funkcional, first and second derivative of a functional, Euler and Lagrange conditions, strong and weak convergence, classic, minimizing and variational formulation of differential problems (examples in mechanics of building structures), numeric solutions to initial and boundary problems, Ritz and Galerkin method, finite-element method, an overview of further variational methods, space and time discretization of evolution problems.

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Requirements for successful completion of the subject are specified by guarantor’s regulation updated for every academic year.

1. Linear, metric, normed, and unitary spaces. Fixed-point theorems.
2. Linear operators, the notion of a functional, special functional spaces
3. Differential operators. Initial and boundary problems in differential equations.
4. First derivative of a functional, potentials of some boundary problems.
5. Second derivative of a functional. Lagrange conditions.
6. Convex functionals, strong and weak convergence.
7. Classic, minimizing and variational formulation of differential problems
8. Primary, dual, and mixed formulation - examples in mechanics of building structures
9. Numeric solutions to initial and boundary problems, discretization schemes.
10. Numeric solutions to boundary problems. Ritz and Galerkin method.
11. Finite-element method, comparison with the method of grids.
12. Kačanov method, method of contraction, method of maximal slope.
13. Numeric solution of general evolution problems. Full discretization and semi-discretization. Method of straight lines. Rothe method of time discretization.
14. An overview of further variational methods: method of boundary elements, method of finite volumes, non-grid approaches. Variational inequalities.

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Rektorys K.: Variační metody v inženýrských problémech a v problémech matematické fyziky. Academia 1999
Nečas J. - Hlaváček I.: Úvod do matematické teorie pružných a pružně plastických těles. SNTL Praha 1983