Course detail

Numerical methods for the variational problems

FAST-DAB036Acad. year: 2023/2024

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.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Entry knowledge

Mathematical and numerical analysis at the level of the course DA61.

Rules for evaluation and completion of the course

Extent and forms are specified by guarantor’s regulation updated for every academic year.

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.

Study aids

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