Basics of Calculus of Variations
FAST-CA058Acad. year: 2019/2020
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.
Institute of Mathematics and Descriptive Geometry (MAT)
Learning outcomes of the course unit
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.
Recommended optional programme components
Recommended or required reading
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Language of instruction
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.
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.
Specification of controlled education, way of implementation and compensation for absences
Extent and forms are specified by guarantor’s regulation updated for every academic year.
Classification of course in study plans