Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Saturday, January 25th, 1997

On the Treatment of Boundary Conditions in Multiscale Methods for Elliptic P.D.E.'s

When using wavelet-based multiscale methods for the numerical solution of elliptic partial differential equations, the satisfaction of boundary conditions is usually a delicate question. After collecting some results on the explicit treatment of boundary conditions by Lagrange multipliers, I would like to make some comments on the efficient and fast iterative solution of the resulting linear system. As it turns out, the use of wavelet bases in a preconditioner for the resulting Schur complement yields uniformly bounded condition numbers. In my talk I will focus on the multiscale bases used for this purpose and their construction, both on the domain and the boundary. Alternatively, a ficticious domain approach might be used for the operator part defined on the domain. I would also like to mention some recent results in this direction employing multiscale bases. Finally, I would like to present some recent numerical examples.