Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation

Abstract

In this work we propose a hybridizable discontinuous Galerkin (hdG) discretization of the high-frequency Helmholtz equation in the presence of point sources and highly heterogeneous and discontinuous wave speed models. We show that it delivers solutions that are provably second-order accurate and do not suffer from the pollution error, as long as a slightly higher order hdG method is used where the polynomial degree is chosen such that $p = \mathcal{O}(\log \omega)$. These results hold even if the discontinuities in the wave speed are not resolved by the hdG mesh, as long as the integration procedure used in the assembly of the stiffness matrix respects the discontinuities. Further, we show that the associated linear systems can be solved using a modification of the method of polarized traces resulting in a method with linear complexity up to a poly-logarithmic factor, or sub-linear complexity in a parallel environment. To our knowledge and surprise, this note contains the first instance of a numerical method that is at the same time fast ($\mathcal{O}(N)$ runtime) and accurate (second-order, pollution-free) in the context of models of geophysical interest.