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Course detail
Numerical Methods III
Course unit code:  FSISN3  

Academic year:  2016/2017  
Type of course unit:  compulsory  
Level of course unit:  Master's (2nd cycle)  
Year of study:  1  
Semester:  winter  
Number of ECTS credits:  3  



























Type of course unit:
Lecture:  26 hours, optionally 

Teacher / Lecturer:  RNDr. Rudolf Hlavička, CSc. 
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. BrambleHilbert'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. 
seminars in computer labs:  13 hours, compulsory 
Teacher / Lecturer:  RNDr. Rudolf Hlavička, CSc. 
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. 12. Programming tools, first introduction. 34. Further details, preparation for writing of the program for solution of an elliptic problem (stationary heat conduction). 56. Developing the program for an elliptic problem. Explanation of the algorithm for the solution of the parabolic problem (nonstationary heat conduction). 78. Developing the program for a parabolic problem. Explanation of the algorithm for the solution of the hyperbolic problem (membrane vibrations). 910. Developing the program for a hyperbolic problem. Explanation of the algorithm for the solution of the eigenvalue problem. 1112. Developing the program for an eigenvalue problem. 13. Teacher's reserve. 