Iterative methods for large sparse systems

Lecture by Dr. Rüdiger Müller
Humboldt-Universität zu Berlin
Summer 2015

Friday: 11:00-13:00
Room: RUD25 - 1.011


  1. Introduction ( additional slides )
    1. Sparse Matrices
    2. Poisson Model Problem
  2. LU Decomposition ( additional slides )
  3. Basic Iterations, Splitting Methods ( additional slides )
    1. Diagonal Methods: Richardson, Jacobi
    2. Additive Triangular Splitting: Gauß-Seidel, SOR
      Matlab programs for illustration: [ test_fp.m ], [ gauss_seidel.m ]
      stiffness matrix is computed using [laplacian.m] by B. C. Smith and A. V. Knyazev
    3. Multiplicative Splitting: ILU
    4. Approximative Inverse
  4. Multigrid Methods ( additional slides )
    1. Multigrid Algorithm
      Matlab programs for illustration: [ test_mg.m ], [ twogrid.m ]
    2. Smoothing Property
    3. Approximation Property, Convergence
  5. Descend Methods, CG-Methods
    1. Gradient Methods
    2. Conjugate Gradient Method (CG)
    3. Preconditioning, PCG-Methods
  6. Block Oriented Methods, Domain Decomposition
    1. Block Methods
    2. Alternating Schwarz-Methods


R. Barrett et al. Templates for the solution of linear systems: building blocks for iterative methods. SIAM, 1987.
W. Hackbusch. Iterative Lösung großer Gleichungssysteme. Teubner, 1991.
Y. Saad. Iterative methods for sparse linear systems. SIAM, 1996.

PDEs and numerical methods

L. C. Evans Partial differential equations. AMS, 1998.
S. Larsson, V. Thomee. Partial differential equations with numerical methods. Springer, 2009.
R. J. LeVeque Finite difference methods for ordinary and partial differential equations SIAM, 2007.

Finite Elements

D. Braess. Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, 2007.
S. Brenner, L.R. Scott. The mathematical theory of finite element methods. Springer, 2008.

J. Alberty, C. Carstensen, S.A. Funken. Remarks around 50 lines of Matlab: short finite element implementation. Numerical Algorithms 20(2-3), 1999.
S.C. Brenner, C. Carstensen. Finite Element Methods. Encyclopedia of Computational Mechanics, Wiley, 2004.
L. Chen. iFEM: an innovative finite element methods package in MATLAB. Preprint, University of Maryland, 2008.
S Funken, D.Praetorius, P. Wissgott. Efficient implementation of adaptive P1-FEM in MATLAB. Comput. Methods Appl. Math. 11, no. 4, 2011.