Boundary value problems on polyhedral domains and applications to numerical methods
Victor Nistor
Abstract:
I will first review some of the issues and earlier results
on boundary value problems on polyhedral domains. The main lesson is
that there is a 'loss of regularity' for boundary value problems on
singular domains if the usual Sobolev spaces are used. This is
inconvenient however in practical applications such as the Finite
Element Method. An alternative approach is to use 'weighted Sobolev'
spaces. Then one can then restore full regularity for elliptic
problems on such domains under the additional condition of that there
are no 'Newmann-Newmann' corners or edges (joint result with Bacuta,
Mazzucato, and Zikatanov). The case of 'Newmann-Newmann' corners
requires some additional ideas. I will then present some results in
this case in two dimensions and how they can be used to construct
quasi-optimal sequences of Finite Element Spaces for transmission and
pure Neumann problems (joint work with Mazzucato and Li). This method
is then generalized to three dimensions.