Course detail
Basics of Calculus of Variations
FAST-NAB018Acad. year: 2024/2025
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
Department
Institute of Mathematics and Descriptive Geometry (MAT)
Entry knowledge
Basic courses of mathematics for bachelor students.
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.
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
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
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
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.