Course detail

Basics of Calculus of Variations

FAST-CA58Acad. year: 2014/2015

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)

Learning outcomes of the course unit

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.

Prerequisites

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

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Techaing methods depend on the type of course unit, as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.

Assesment methods and criteria linked to learning outcomes

Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops. The result of the semester examination is given by the sum of maximum of 70 points obtained for a written test and a maximum of 30 points from the seminar.

Course curriculum

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.

Work placements

Not applicable.

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.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Nečas J. - Hlaváček I.: Úvod do matematické teorie pružných a pružně plastických těles. SNTL Praha, 1983. (CS)
Rektorys K.: Variační metody v inženýrských problémech a v problémech matematické fyziky. Academia, 1999. (CS)
S. Fučík, A. Kufner: Nonlinear Differential Equations. Elsevier, 1980. (EN)

Classification of course in study plans

  • Programme N-K-C-SI Master's

    branch V , 1 year of study, summer semester, elective

  • Programme N-P-C-SI Master's

    branch V , 1 year of study, summer semester, elective

  • Programme N-P-E-SI Master's

    branch V , 1 year of study, summer semester, elective

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.