Per Olsson
Microsoft, Commercial Software Engineering
This presentation focuses on projections as a technique for implementing boundary conditions that appear in the context of Initial-Boundary Value Problems (IBVP). These projections are inextricably linked to summation by parts in that they are built on the same norms - or scalar products - required by difference operators that exhibit a summation by parts property. After a brief summary of the pioneering work by Kreiss and Scherer he will give an overview of his previous research related to projections and stability, including a possible approach to extending projections to nonlinear stability (entropy stability).
The main part of the talk focuses on a novel approach to implementing the projection operators using pseudoinverses in the sense of Penrose. His characterization of pseudoinverses will be extended to general finite-dimensional inner product spaces. Based on this generalization it is possible to implement the boundary projection as P = I - (L^+)L, where L^+ is the pseudoinverse of the boundary operator L.