Course detail

# Numerical methods for the variational problems

FAST-DA66Acad. year: 2020/2021

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.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

Basic notions of linear algebra and mathematical analysis, elementary methods for exact solutions of differential equations, methods for approximate solutions of systems of linear and non-linear equations, interpolation and approximation of functions, numerical differentiation and numerical integration.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Functional and its Euler equation, the simlest problem ov 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.

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.

Classification of course in study plans

• Programme D-P-C-SI (N) Doctoral

branch PST , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-C-SI (N) Doctoral

branch PST , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-E-SI (N) Doctoral

branch PST , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-E-SI (N) Doctoral

branch PST , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-E-SI (N) Doctoral

branch MGS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-E-SI (N) Doctoral

branch MGS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-C-SI (N) Doctoral

branch KDS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-C-SI (N) Doctoral

branch KDS , 2. year of study, winter semester, 10 credits, compulsory-optional
branch MGS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-C-SI (N) Doctoral

branch MGS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-E-SI (N) Doctoral

branch FMI , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-C-SI (N) Doctoral

branch FMI , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-E-SI (N) Doctoral

branch FMI , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-C-SI (N) Doctoral

branch FMI , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-E-SI (N) Doctoral

branch KDS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-E-SI (N) Doctoral

branch KDS , 2. year of study, winter semester, 10 credits, compulsory-optional
branch VHS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-E-SI (N) Doctoral

branch VHS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-C-SI (N) Doctoral

branch VHS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-C-SI (N) Doctoral

branch VHS , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-K-C-GK Doctoral

branch GAK , 2. year of study, winter semester, 10 credits, compulsory-optional

• Programme D-P-C-GK Doctoral

branch GAK , 2. year of study, winter semester, 10 credits, compulsory-optional

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer