Course Details

Basics of Calculus of Variations

Academic Year 2024/25

NAB018 course is part of 1 study plan

​ NPC-SIV Summer Semester 1st year

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

Credits

5 credits

Language of instruction

Czech

Semester

summer

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Forms and criteria of assessment

course-unit credit and examination

Entry Knowledge

Basic courses of mathematics for bachelor students.

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.

Basic Literature

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

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/282733

Lecture

13 weeks, 2 hours/week, elective

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

13 weeks, 2 hours/week, compulsory

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.