Finite Element Method in Applications

Garant předmětu:

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).

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.

Application of the finite element method: heat conduction, wave equation, convection-diffusion problem, linear elasticity problem, Stokes and Navier-Stokes problem.

Syllabus of tutorials:

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).

Study materials:

[1] P. Sváček and M. Feistauer. Metoda konečných prvků. Vydavatelství ČVUT, Praha, 2006.

[2] C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, 1992.

[3] K. Rektorys. Variační metody. Academia, Prague, 1999

[4] E. Vitásek. Základy teorie numerických metod pro řešení diferenciálních rovnic. Academia, Prague, 1994

[5] K. Rektorys. Variational Methods in Mathematics, Science and Engineering. Reidel, Dordrecht, Holland, 1980

[6] P. G. Ciarlet. The Finite Element Methods for Elliptic Problems. North-Holland Publishing, 1979

[7] A. K. Aziz. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, 1972.

[8] I. Babus̆ka, T. Strouboulis, The Finite Element Method and Its Reliability, Clarendon Press, 2001.

[9] B. Szabó, I. Babuška, Finite Element Analysis, John Wiley & Sons, 12. 4. 1991.

Note:

Further information:

http://marian.fsik.cvut.cz/~svacek/fem/

No time-table has been prepared for this course

The course is a part of the following study plans:

• 13 136 NSTI MMT 2012 základ (compulsory course in the program)