Course Details

Basics of Calculus of Variations

NAB018 course is part of 1 study plan

NPC-SIV Summer Semester 1st year

Course Guarantor

Institute

Language of instruction

Czech

Credits

5 credits

Semester

summer

Forms and criteria of assessment

course-unit credit and examination

Offered to foreign students

Not to offer

Course on BUT site

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.