Numerical Analysis Seminar

In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree p on each mesh element. Earlier works showed that there is a \trial-to-test" operator T, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator T is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply T. In practical computations, T is approximated using polynomials of some degree r > p on each mesh element. We show that this approximation maintains optimal convergence rates, provided that r p + N, where N is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.

Fast computation of drum modes using the spectrum of the Neumann-to-Dirichlet map We present and analyze a new method for numerical computation of the spectrum and eigenfunctions of a planar star-shaped domain with Dirichlet boundary condition. The method is 'fast' since it is computes a cluster of eigenfunctions (numbering of order the square-root of the eigenvalue) in the time usually taken to compute a single one. In practice, with 400 wavelengths across the domain, and relative error 1e-10, this speed-up is around 1e3. It is related to the little-understood 'scaling method', but, in contrast, has a rigorous error analysis and allows higher-order accuracy. We will include some applications to quantum chaos. Joint work with Andrew Hassell (ANU).