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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2019/2020

Finite Element Method

The course is not on the list Without time-table
Code Completion Credits Range
2016110 Z 3 2P+1C
Lecturer:
Tutor:
Supervisor:
Department of Technical Mathematics
Synopsis:

Mathematical background of the finite element method. Banach and Hilbert spaces. Linear forms, bilinear forms, scalar product. Hölder and Cauchy inequality. Lax-Milgram theorem. Lebesgue and Sobolev spaces. Sobolev imbeddings theorem and the trace theorem. Green theorem. Substitution theorem. Poincare-Friedrichs inequality.

Basic principle of the finite element method. Example of application for 1D problem, classical and weak solution, error estimates. Abstract variational formulation, Ritz and Galerkin problem. Existence and uniquness of the solution. Discrete Ritz and Galerkin problems. Cea's lemma (error estimate).

Application of finite element method for 2D problem. Weak formulation for the case of zero Dirichlet boundary condition. Discretization using Lagrangiang linear elements, finite element space and base construction. Assembling of the stiffness matrix and the load vector. Weak formulation for mixed boundary conditions. Reference element and mapping, extension to 3D and higher order finite elements.

Solution of the discrete problem – systems of linear equations. Direct and iterative methods. Gradient methods, conjugate gradient method and preconditioning.

Requirements:
Syllabus of lectures:

Mathematical background of the finite element method. Banach and Hilbert spaces. Linear forms, bilinear forms, scalar product. Hölder and Cauchy inequality. Lax-Milgram theorem. Lebesgue and Sobolev spaces. Sobolev imbeddings theorem and the trace theorem. Green theorem. Substitution theorem. Poincare-Friedrichs inequality.

Basic principle of the finite element method. Example of application for 1D problem, classical and weak solution, error estimates. Abstract variational formulation, Ritz and Galerkin problem. Existence and uniquness of the solution. Discrete Ritz and Galerkin problems. Cea's lemma (error estimate).

Application of finite element method for 2D problem. Weak formulation for the case of zero Dirichlet boundary condition. Discretization using Lagrangiang linear elements, finite element space and base construction. Assembling of the stiffness matrix and the load vector. Weak formulation for mixed boundary conditions. Reference element and mapping, extension to 3D and higher order finite elements.

Solution of the discrete problem – systems of linear equations. Direct and iterative methods. Gradient methods, conjugate gradient method and preconditioning.

Syllabus of tutorials:

Basic principle. Mathematical background. Weak formulation of a boundary value problem. Approximation theory. Variational methods. Systems of linear equations. Heat transfer problem. Wave problem. Fluid mechanics.

Study Objective:

Mathematical background of the finite element method. Banach and Hilbert spaces. Linear forms, bilinear forms, scalar product. Hölder and Cauchy inequality. Lax-Milgram theorem. Lebesgue and Sobolev spaces. Sobolev imbeddings theorem and the trace theorem. Green theorem. Substitution theorem. Poincare-Friedrichs inequality.

Basic principle of the finite element method. Example of application for 1D problem, classical and weak solution, error estimates. Abstract variational formulation, Ritz and Galerkin problem. Existence and uniquness of the solution. Discrete Ritz and Galerkin problems. Cea's lemma (error estimate).

Application of finite element method for 2D problem. Weak formulation for the case of zero Dirichlet boundary condition. Discretization using Lagrangiang linear elements, finite element space and base construction. Assembling of the stiffness matrix and the load vector. Weak formulation for mixed boundary conditions. Reference element and mapping, extension to 3D and higher order finite elements.

Solution of the discrete problem – systems of linear equations. Direct and iterative methods. Gradient methods, conjugate gradient method and preconditioning.

Study materials:

[1] C.Johnson: Numerical Solution of Partial Differential Equation by the Finite Element Method, Cambridge University Press, 1987, ISBN 0-521-34758-0.

Note:
Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2019-12-15
For updated information see http://bilakniha.cvut.cz/en/predmet10730902.html