Course detail

Numerical methods for the variational problems

FAST-DAB036Acad. year: 2021/2022

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

prof. Ing. Jiří Vala, CSc.

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.

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 DPA-M Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPC-M Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPA-K Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPC-K Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPA-V Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPC-V Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPA-S Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPC-S Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPA-E Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPC-E Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPA-GK Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DPC-GK Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKA-M Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKC-M Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKA-K Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKC-K Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKA-V Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKC-V Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKA-S Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKC-S Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKA-E Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKC-E Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKA-GK Doctoral 2 year of study, winter semester, compulsory-optional
  • Programme DKC-GK Doctoral 2 year of study, winter semester, compulsory-optional

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

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.