Domain decomposition methods for partial differential equations

Description

Many approximation methods for partial differential equations reduce the problem under consideration to a large linear or non--linear system of equations. In large scale simulations, the number of unknowns may easily reach tens of million and it is often required to move to massively parallel computers. In addition, the efficient solution of very large linear systems requires some kind of preconditioning. Domain decomposition methods provide powerful preconditioners for systems arising from the approximation of partial differential equations and can quite naturally be implemented on parallel computers. In a domain decomposition approach, a preconditioner is obtained by solving problems over smaller subregions and then patching together the local solutions, by imposing continuity of suitable traces on the interfaces between the subregions. A coarse problem is usually added in order to obtain a scalable method.

Domain decomposition methods are also employed as approximation methods. A physical object or region can be naturally partitioned into subregions with different properties and different and independent approximation methods might be advisable in different regions (finite elements, spectral elements, ...).
The analysis of domain decomposition methods is often not trivial and requires the use of fairly sophisticated mathematical tools.

The following topics are available for a semesterarbeit or a Diploma thesis on the implementation and/or the analysis of domain decomposition methods:

• Iterative substructuring preconditioners for convection dominated problems;
• Approximations on non matching grids of flow problems;
• Approximations on non matching grids of rotating geometries;
• Preconditioners for scattering problems at high frequency;
• Preconditioners for hp finite element approximations.

Contacts

Dr. Andrea Toselli.