Lecturer(s)


Netuka Horymír, RNDr. Ph.D.

Ligurský Tomáš, RNDr. Ph.D.

Course content

1. Recapitulation: Sobolev spaces and their main properties, variational formulations of elliptic boundary value problems, RitzGalerkin method. 2. FEM as a RitzGalerkin method. Triangulation and its principles. Basis functions selection in the FEM. Comparison to classical procedures. 3. Solution of the 1D problems using the finite element method: mixed boundary value problem of the 2nd order by means of linear elements. 4. Definition of a finite element. Punisolvency. Local basis functions. Typical examples for finite elements of Lagrange's and Hermite's type. Finite element spaces. 5. Introduction to convergence theory of the FEM. Cea's lemma. BrambleHilbert lemma. Reference elements. Analysis of order of convergence for linear triangular elements. 6. Approximation theory in Sobolev spaces with results for finite element convergence theory. Error estimates for approximate solution in H1norm and in L2norm. 7. Quadrature formulas used in FEM. Affine equivalent elements. Effect of numerical integration to finite element solution. 8. FEM in domains with nonpolygonal boundary. Concept of isoparametric finite elements. Nonconforming FEM. 9. Solution of the 1D problems using the finite element method: beam bending problem by means of cubic Hermite elements. 10. FEM algorithm. Assembling techniques for finite element systems of equations. Methods to solve these systems Cholesky method and conjugate gradient method in the case of large sparse systems of equations. Postprocessing. 11. Finite element approximation of the solution to parabolic problems. Weak formulation of parabolic problems. Galerkin method for time dependent problems. Principle of the Rothe's method. 12. Fundamentals of mixed finite element method. Introduction to solution of nonlinear problems using FEM. Types of nonlinearities. Examples: nonlinear beam model, contact problems.

Learning activities and teaching methods

Lecture, Monologic Lecture(Interpretation, Training)
 Attendace
 39 hours per semester
 Preparation for the Exam
 50 hours per semester
 Semestral Work
 35 hours per semester

Learning outcomes

Understand and be able to use the most widespread computational method for boundary value problems.
Knowledge Gain knowledge about wellknown method for solution of boundary value problems.

Prerequisites

Student has to pass the course Variational methods (KMA/VM). Standard knowledge from numerical mathematics.
KMA/VM

Assessment methods and criteria

Oral exam, Seminar Work
Credit: the student has to compute a given example. Exam: the student has to understand the subject and be acquainted with theoretical and practical aspects of the method.

Recommended literature


J. Haslinger. (1980). Metoda konečných prvků pro řešení eliptických rovnic a nerovnic. MFF UK, Praha.

J. N. Reddy. (1993). An Introduction to the Finite Element Method, 2nd edition. McGrawHill New York.

M. Křížek, P. Neittaanmaki. (1990). Finite element approximation of variational problems and applications. Longman.

P. Ciarlet. (1978). The Finite Element Method for Elliptic Problems. NorthHolland Amsterdam.

S.C. Brenner, L.R. Scott. (2008). The Mathematical Theory of Finite Element Methods. Springer.
