Course detail

Basics of Calculus of Variations

FAST-NAB018Acad. year: 2022/2023

Intorduction to variational methods, applications to the analysis of differential equations.

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

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.

Prerequisites

Basic courses of mathematics for bachelor students.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

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

BOUCHALA J.: Variační metody. VŠB-TU Ostrava 2012 (CS)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme NPC-SIV Master's 1 year of study, summer semester, compulsory-optional

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.