263-5001-00L Introduction to finite elements and sparse linear system solving
Fall semester 2010

Friday 9-12, CAB G56
4 ECTS credits

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 equations that are typical for the FE method. We will consider both direct and iterative methods.

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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).

Literature

1. H. Elman, D. Sylvester, A. Wathen: Finite elements and fast iterative solvers.   Oxford University Press,  Oxford,  2005.
2. P.Knabner, L.Angermann: Numerik partieller Differentialgleichungen. Springer, Berlin, 2000. Engl. translation: Numerical methods for elliptic and parabolic partial differential equations. Springer, New York, 2003.
3. H.R. Schwarz: Die Methode der finiten Elemente (3rd ed.). Teubner,  Stuttgart,  1991.
4. Y. Saad:Iterative methods for sparse linear systems (2nd ed.). SIAM,  Philadelphia, 2003.
5. T. Davis: Direct Methods for Sparse Linear Systems. SIAM,  Philadelphia, 2006.
6. I.S. Duff, A. M. Erisman: Direct Methods for Sparse Matrices. Oxford University Press,  Oxford,  1986.

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