Conditioning of Finite Element Equations with Anisotropic Meshes
Dienstag, den 27.09.2011, 16.15 Uhr in MA 313 Abstract:

In n >= 2 dimensions, it has been proven by Bank and Scott that the condition number of finite element equations does not degrade significantly on adaptive meshes if the mesh remains locally quasi-uniform and an appropriate scaling of the resulting system is used.

In this talk, I will present a generalization of the result by Bank and Scott. The developed bound on the condition number is valid for general meshes, without any assumptions on the shape of mesh elements. As in isotropic case, an appropriately chosen, mesh-dependent diagonal scaling can be used to significantly improve the conditioning of the resulting linear system.

An interesting result is that the bound on the condition number of the scaled system is mostly the same as for the uniform case even if the mesh contains highly anisotropic elements, provided the number of anisotropic elements is relatively small. A similar result is also achieved for n=1 dimension.

Diagonal stability and completely positive matrices
Dienstag, den 30.08.2011, 16.15 Uhr in MA 313 Abstract:

The talk will consist of a short survey of the theory of completely positive matrices and relate it to a newly defined concept of a common diagonal Lyapunov matrix. A necessary and sufficient condition for the existence of such a matrix will be derived. The second part of the talk is based on a recent paper with C. King and R. Shorten.