Course detail
Numerical methods for the variational problems
FAST-DAB036Acad. year: 2020/2021
Introduction to the variatoinal calculus, analysis of initial and boundary problems for ordinary and partial differential equations, selected applications to civil engineering.
Language of instruction
Czech
Number of ECTS credits
10
Mode of study
Not applicable.
Guarantor
Department
Institute of Mathematics and Descriptive Geometry (MAT)
Learning outcomes of the course unit
Not applicable.
Prerequisites
Mathematical and numerical analysis at the level of the course DA61.
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. Functional and its Euler equation, the simlest problem of calculus of variations.
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.
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
Not applicable.
Aims
Basics of calculus of variations, numerical methods for variationally formulated differential boundary-value problems. The studied boudary-value problems are mathematical models of processes often occuring in the practice of civil engineers.
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
Not applicable.
Recommended reading
Not applicable.
Classification of course in study plans
- Programme DPC-S Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-M Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-K Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-V Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-E Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-GK Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPA-E Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPA-GK Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPA-K Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPA-M Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPA-S Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPA-V Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-E Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-GK Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-K Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-M Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-S Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DPC-V Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DKA-E Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DKA-GK Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DKA-K Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DKA-M Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DKA-S Doctoral 2 year of study, winter semester, compulsory-optional
- Programme DKA-V Doctoral 2 year of study, winter semester, compulsory-optional
Type of course unit
Lecture
39 hod., optionally
Teacher / Lecturer
Syllabus
1. Functional and its Euler equation, the simlest problem of calculus of variations.
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.