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263-5001-00L 5 Credits BSC , DS , MSC , WBZ D-MATH , D-INFK

Introduction to finite elements and sparse linear system solving

Lecturers & Examiners: Prof. em. Peter Arbenz, Christof Vömel
VVZ CR n/a

Last Updated: 2026-02-05 15:24:42

Abstract

The finite element (FE) method is the method of choice for (approximately) solving partial differential equations on complicated domains. In the first third of the lecture, we give an introduction to the method. The rest of the lecture will be devoted to methods for solving the large sparse linear systems of equation that a typical for the FE method. We will consider direct and iterative methods

Content

I. THE FINITE ELEMENT METHOD (1) Introduction, model problems. (2) Variational formulations. (3) Galerkin finite element method. (4) Implementation aspects. II. DIRECT SOLUTION METHODS (5) LU and Cholesky decomposition. (6) Sparse matrices. (7) Fill-reducing orderings. III. ITERATIVE SOLUTION METHODS (8) Stationary iterative methods, preconditioning. (9) Preconditioned conjugate gradient method (PCG). (10) Incomplete factorization preconditioning. (11) Multigrid preconditioning. (12) Nonsymmetric problems (GMRES, BiCGstab). (13) Indefinite problems (SYMMLQ, MINRES).

Resources

Literature

[1] H. Elman, D. Sylvester, A. Wathen: Finite elements and fast iterative solvers. OUP, Oxford, 2005. [2] Y. Saad: Iterative methods for sparse linear systems (2nd ed.). SIAM, Philadelphia, 2003. [3] T. Davis: Direct Methods for Sparse Linear Systems. SIAM, Philadelphia, 2006. [4] C.W. Oosterlee, C. Vuik: Scientific Computing. Lecture notes, TU Delft, 2007. [5] H.R. Schwarz: Die Methode der finiten Elemente (3rd ed.). Teubner, Stuttgart, 1991.

General Information

Language
English
Levels
BSC , DS , MSC , WBZ
Frequency
Yearly recurring

Examination

Type
end-of-semester examination
Oral semester end exam. Duration: 30 min. per candidate

Course Components

Type Title Time & Place Hours
lecture Introduction to finite elements and sparse linear system solving
  • Fri 09:15-11:00 (CAB H 52)
2 h weekly
exercise Introduction to finite elements and sparse linear system solving
  • Fri 11:15-12:00 (CAB H 52)
1 h weekly

Offered In