Course detail

# Numerical Methods III

FSI-SN3Acad. year: 2020/2021

The course deals with the mathematical foundations of the finite element method and with the explanation of selected finite elements algorithms of basic engineering problems. First, the fundamentals of the theory of Sobolev spaces is presented. Using it, the term of the weak solution of a boundary value problem of an elliptic partial differential equation is explained. This weak solution is then approximated by the finite element method in various ways. Various types of finite elements are introduced. Also discussed is the theory of interpolation and numerical integration in the finite element method. The convergence of the finite element method is analysed. Using a linear triangular finite element, they will assemble and debug their own programs for the solution of elliptic, parabolic, hyperbolic and eigenvalue problems.

Supervisor

Department

Learning outcomes of the course unit

The theory of Sobolev spaces, the theory of interpolation and the theory of numerical integration on finite elements are the basic mathematical tools in the finite element method. The programming of algorithms based on the linear triangular element is the starting point for understanding more sophisticated finite element implementation techniques.

Prerequisites

Differential and integral calculus for multivariable functions. Fundamentals of functional analysis. Partial differential equations. Numerical methods, especially interpolation, quadrature and solution of systems of ODE. Programming in the Matlab and Visual Studio enviroment.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

A. Ern, J.-L. Guermond: Theory and Practice of Finite Elements, Springer Series in Applied Mathematical Sciences, Vol. 159 (2004) 530 p., Springer-Verlag, New York (EN)

A. Ženíšek: Matematické základy metody konečných prvků, [on-line], available from:
http://mathonline.fme.vutbr.cz/Numericke-metody-III/sc-1151-sr-1-a-142/default.aspx. (CS)

S.C. Brenner, L.R. Scott: The Mathematical Theory of Finite Element Methods, Springer-Verlag, 2002. (EN)

L. Čermák: Algoritmy metody konečných prvků, [on-line], available from:
http://mathonline.fme.vutbr.cz/Numericke-metody-III/sc-1151-sr-1-a-142/default.aspx. (CS)

P. Knabner, L. Angermann: Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, 2003. (EN)

C. Jonson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1995. (EN)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Graded course-unit credit is awarded on the following conditions: 30% on weekly programming assignments, 70% on the individual project. Participation in the lessons may be reflected in the final mark. If we measure the exam success in percentage points, then the classification grades are: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The aim of the course is to familiarise students with the basis of the finite element method and to equip them with the ability to study papers and books in this field and in related branches. The will be also acquainted with algorithmisation practices and standard programming techniques routinely used in finite element implementations.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is required. Lessons are planned according to the week schedules. Absence from lessons may be compensated by the agreement with the teacher supervising the seminars.

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

The first four lectures will be devoted to the explanation of the algorithm for solution of the model problem of type "stationary heat conduction" in a plane polygonal domain using linear triangular finite elements. This enables students from the very beginning of practicals to start experimenting with code programming. Only the following lectures will concentrate on the mathematical theory of finite elements.

1. Classical and variational formulation, triangulation, piecewise linear functions.

2. Discrete variational formulation, elementary matrices and vectors.

3. Elementary matrices and vectors - continuation.

4. Assembly of global system of equations, its solution, postprocessing.

5. Selected pieces of knowledge of functional analysis. The space W^k_2.

6. Traces of functions from the space W^k_2. Friedrich's and Poincare's inequality.

7. Bramble-Hilbert's lemma. Sobolev's imbedding theorem.

8. Formal equivalence of the elliptic boundary value problem and the related variational problem.

9. Finite element spaces of Lagrange's type. Definition of approximate solution. Existence and uniqueness theorem.

10. Transformation of a general triangle onto the reference triangle. Relations between norms on the general triangle and on the reference triangle.

11. Interpolation theorem.

12. Numerical integration.

13. Adaptivity in FEM.

Computer-assisted exercise

13 hours, compulsory

Teacher / Lecturer

Syllabus

Practicals will take place in a computer lab with the support of the MATLAB and Visual Studio. The algorithm for the elliptic problem will be explained during the first four lessons. The algorithms for the parabolic, hyperbolic and eigenvalue problems will be explained in brief on practicals. It is supposed that students will work individually with lecture notes (containing detailed descriptions of algorithms). Students are also expected to create and debug individually their own MATLAB programs.

1-2. Programming tools, first introduction.

3-4. Further details, preparation for writing of the program for solution of an elliptic problem (stationary heat conduction).

5-6. Developing the program for an elliptic problem. Explanation of the algorithm for the solution of the parabolic problem (nonstationary heat conduction).

7-8. Developing the program for a parabolic problem. Explanation of the algorithm for the solution of the hyperbolic problem (membrane vibrations).

9-10. Developing the program for a hyperbolic problem. Explanation of the algorithm for the solution of the eigenvalue problem.

11-12. Developing the program for an eigenvalue problem.

13. Teacher's reserve.

eLearning

**eLearning:** currently opened course