Course detail
Numerical methods for the variational problems
FAST-DA66Acad. year: 2022/2023
1. Introduction to the variatoinal calculus: Examples of functionals, the simplest problem of variational calculus, Euler equation of a functional.
2. Differential problems: Classical and variational formulations of boundary-value differential problems. Discretization of stationary differential problems by the finite-difference, Galerkin Ritz methods. Standard time-discretizations of non-stationary differential problems.
3. Formulation and numerical solution of the heat-conduction problem, the linear elasticity problem, of the linear flow problems, of the Navier-Stokes equations and of selected models of simultaneous moisture and heat distribution in porous media.
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Learning outcomes of the course unit
Prerequisites
Co-requisites
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Course curriculum
2. Concrete examples of functionals and related Euler equations. Elementary solutions.
3. Derivation of an elliptic problem for ODE of degree 2, the problems of heat conduction and distribution of polution.
4. Discretization of the elliptic problem for ODE of degree 2 by the standard finite difference method, stability of numerical solutions.
5. Variational (weak) and minimization formulation of the elliptic problem for the elliptic problem for ODE of degree 2.
6. The Ritz and Galerkin methods.
7. Discretization of the elliptic problem for ODE of degree 2 by the finite element method.
8. Discretization of the variational formulation of the elliptic problem for ODE of degree 2 by the finite element method.
9. Discretization of the minimization formulation of the elliptic problem for ODE of degree 2 by the finite element method.
10. Discretization of the variational formulation of the elliptic problem for PDE of degree 2 by the finite element method.
11. Variational formulation and the finite element method for the linear elasticity problem.
12. Navier-Stokes equations and their numerical solution by the particle method.
13. A mathematical model of simultaneous distribution of moisture and heat in porous materials, discretizations.
Work placements
Aims
Specification of controlled education, way of implementation and compensation for absences
Recommended optional programme components
Prerequisites and corequisites
Basic literature
BOUCHALA J.: Variační metody. VŠB-TU Ostrava 2012. (CS)
Recommended reading
Classification of course in study plans
- Programme D-K-E-CE (N) Doctoral
branch FMI , 2 year of study, winter semester, compulsory-optional
branch KDS , 2 year of study, winter semester, compulsory-optional
branch MGS , 2 year of study, winter semester, compulsory-optional
branch VHS , 2 year of study, winter semester, compulsory-optional
branch PST , 2 year of study, winter semester, compulsory-optional - Programme D-K-C-SI (N) Doctoral
branch VHS , 2 year of study, winter semester, compulsory-optional
branch MGS , 2 year of study, winter semester, compulsory-optional
branch PST , 2 year of study, winter semester, compulsory-optional
branch FMI , 2 year of study, winter semester, compulsory-optional
branch KDS , 2 year of study, winter semester, compulsory-optional - Programme D-P-E-CE (N) Doctoral
branch PST , 2 year of study, winter semester, compulsory-optional
branch FMI , 2 year of study, winter semester, compulsory-optional
branch KDS , 2 year of study, winter semester, compulsory-optional
branch MGS , 2 year of study, winter semester, compulsory-optional
branch VHS , 2 year of study, winter semester, compulsory-optional - Programme D-P-C-SI (N) Doctoral
branch PST , 2 year of study, winter semester, compulsory-optional
branch FMI , 2 year of study, winter semester, compulsory-optional
branch KDS , 2 year of study, winter semester, compulsory-optional
branch MGS , 2 year of study, winter semester, compulsory-optional
branch VHS , 2 year of study, winter semester, compulsory-optional - Programme D-P-C-GK Doctoral
branch GAK , 2 year of study, winter semester, compulsory-optional
- Programme D-K-C-GK Doctoral
branch GAK , 2 year of study, winter semester, compulsory-optional
Type of course unit
Lecture
Teacher / Lecturer
Syllabus