Course detail

Basics of Calculus of Variations

FAST-CA058Acad. year: 2024/2025

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.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. Ing. Jiří Vala, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Entry knowledge

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

Rules for evaluation and completion of the course

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

Aims

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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Not applicable.

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

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.

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

Follows directly particular lectures. 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.