In fact, a second element in our theory consists of simple recursive formulas which allow us to obtain the power series by considering a sequence of easily solvable scattering problems. The analytic extension of the fields to large perturbations can then be achieved via Pade approximation. This is joint work with Oscar Bruno, Applied Mathematics, Caltech. A fully discrete hp-scheme is discussed exemplarily for the heat equation. The use of certain mesh-design principles for spatial discretizations yields exponential rates of convergence in time and space.
Numerical examples confirm the theoretical results.
Previous work on analyticity of DNO has produced formulae which, as we have found, are very ill-conditioned. We show how a simple change of variables can lead to recursions which satisfy analyticity estimates without relying on subtle cancellation properties at the heart of previous formulae. I will conclude with recent numerical results of the performance of a scheme based on our approach. Standard formulations for Magnetostatics strongly enforce the constitutive law and one of Maxwell's equations, while weakly satisfying the other one. Due to the usually poorly indentified material properties, it would be preferable to impose both Maxwell's equations strongly, while approximating the constitutive law.