Course: Finite Element Method

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Course title Finite Element Method
Course code KMA/MKP
Organizational form of instruction Lecture + Exercise
Level of course Master
Year of study not specified
Semester Winter
Number of ECTS credits 4
Language of instruction Czech
Status of course Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
  • 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, Ritz-Galerkin method. 2. FEM as a Ritz-Galerkin 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. P-unisolvency. 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. Bramble-Hilbert 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 H1-norm and in L2-norm. 7. Quadrature formulas used in FEM. Affine equivalent elements. Effect of numerical integration to finite element solution. 8. FEM in domains with non-polygonal 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 well-known method for solution of boundary value problems.
Student has to pass the course Variational methods (KMA/VM). Standard knowledge from numerical mathematics.

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. McGraw-Hill 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. North-Holland Amsterdam.
  • S.C. Brenner, L.R. Scott. (2008). The Mathematical Theory of Finite Element Methods. Springer.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science General Physics and Mathematical Physics (2014) Physics courses 2 Winter
Faculty of Science Applied Mathematics (2014) Mathematics courses 2 Winter