Spezielle Numerische Methoden - Randelementmethoden SS 2013
Hinweis: Diese Vorlesung wird auf Englisch abgehalten
Lecturer: Dr. Clemens Pechstein
Time: Thursday, 13.45 – 15.15
Room: S2 054
In special cases, partial differential equations can be reformulated as integral equations, which live only on the boundary of the computational domain. To derive these boundary integral equations, one needs the fundamental solution, theory of distributions, and trace operators. The boundary element method (BEM) is now a special kind of finite element method to discretize the integral equations. Opposed to standard FEM, the unknowns only live on the boundary. The BEM is very suitable for exterior problems, or any other problems where a volume discretization is difficult/costy.
General information
Contents
- Formulation of (elliptic) PDEs as boundary integral equations
- Properties of these equations and the underlying boundary integral operators
- Numerical methods: Collocation and projection (Galerkin) methods, a priori error estimates
Further topics (depends on the time)
- fast BEM: one drawback of standard BEM is, that the system matrices are usually dense (not sparse). For large problems, more sophisticated techniques have to be used to obtain more efficient solvers.
- BEM-FEM coupling: how to combine FEM- and BEM-discretizations (in order to exploit the advantages of both methods)
Required Knowledge
- Partial Differential Equations
- Numerical Methods for Partial Differential Equations
- Ideal (but not required): Numerical Methods for Elliptic Partial Differential Equations, Integral Equations
Lecture Notes
Examination
- Hand in 10 exercises (out of 26) before end of June; if you hand in earlier, I can give feedback.
- Plus small written exam with general questions on Friday, June 21, 10.00, Room 0346 in Science Park 2, 3rd floor
Literature
-
Steinbach O.:
Numerical Approximation Methods for Elliptic Boundary
Value Problems - Finite and Boundary Elements.
Springer, New York, 2008.
Also available in German. A very intuitive book, ideal for beginners. Covers boundary integral equations and their discretization including the basics of fast BEM. -
McLean W.:
Strongly Elliptic Systems and Boundary Integral Equations.
Cambridge University Press, Cambridge, 2000.
My personal favourite. The theory is general and very exact. However, this book does not cover discretization. -
Sauter S., Schwab C.:
Boundary Element Methods.
Springer, Berlin, 2011.
Also available in German. More detailed and more exact/formal than Steinbach's book, but also more difficult to read.
-
Rjasanow S., Steinbach O.:
The Fast Solution of Boundary Integral Equations
- Mathematical and Analytical Techniques with Applications to Engineering.
Springer, New York, 2007.
A book written for engineers: it is explained how the methods work, but not always why. -
Bebendorf, M.:
Hierarchical Matrices - A Means to Efficiently Solve Elliptic Boundary Value Problems.
Vol. 63 of Lecture Notes in Computational Science and Engineering, Springer, Berlin, Heidelberg, 2008.
A special book on hierarchical matrices and ACA (one possibility to get fast BEM).
top
