Dr. Matthias Taus, Postdoctoral Associate working with Prof. Laurent Demanet in the MIT Department of Mathematics, presents "A fast and high-order Hybridizable Discontinuous Galerkin solver for the high-frequency Helmholtz equation" at the MIT Earth Resources Lab on April 15, 2016.

"Solving the time-harmonic wave equation for heterogeneous wave-speeds in the high-frequency regime is a ubiquitous problem in geophysical exploration but it is still open in the context of numerical analysis from the point of view of both optimal accuracy and optimal complexity. The first documented algorithm with truly scalable complexity (i.e. with a runtime sublinear in the number of volume unknowns in a parallel environment) for the high-frequency Helmholtz equation is the method of polarized traces. However, previous versions of this method were either restricted to a low order of accuracy, or suffered from computationally unfavorable complexity in the p-th order case. In this talk, we rectify this issue by proposing a high-order method of polarized traces based on a primal Hybridizable Discontinuous Galerkin (HDG) discretization in a domain decomposition setting. Each stage of the algorithm is either embarrassingly parallel or has a linear computational complexity with respect to the number of degrees of freedom, independently of the order p. This is an important improvement because, especially in the high-frequency regime, high-order discretizations are required to attenuate the pollution error, even in settings when the medium is not smooth. In addition, HDG is a welcome upgrade for the method of polarized traces, since it can be made to work with flexible meshes that align with discontinuities in heterogeneous wave-speeds, and it allows for adaptive refinement in h and p. We provide some examples to corroborate the optimal convergence and complexity claims for problems involving heterogeneous discontinuous wave-speeds. "