## Department of Mathematics, Applied Mathematics and Statistics

**Monday, March 3, 2014 **(3:30-4:30 p.m., Yost 306)

**Title: An Invariant Imbedding Approach **

to Domain Decomposition

**Speaker:**Joseph Volzer (Case Western Reserve University)

**Abstract: **We consider the problem of numerically solving the wave scattering problem in two dimensions, when the scatterer consists of an impenetrable sound-soft scatterer surrounded by a compactly supported penetrable scattering medium. The scattering problem in the exterior domain is solved using boundary integral equations and spectral methods, while the solution near the scatterer is best treated by finite element methods. It is well known that these solutions can be glued together using non-reflecting boundary conditions, a common choice being the Dirichlet-to-Neumman (Steklov-PoincarĂ©) map. If the support of the scattering medium is large, the interior problem may require a large mesh and become computationally intense. We consider an alternative method based on the idea of invariant embedding: first numerically solve for the DtN map on a boundary of a small domain merely enclosing the sound-soft scatterer, and then radially propagate the map out of the support of the scattering medium. Special attention needs to be paid to the stability of the propagation scheme that requires a solution of a matrix-valued Riccati equation. It is shown that by applying an appropriate Cayley transform on the matrix, problems arising from resonances can be avoided.