A short note on a fast and high-order hybridizable discontinuous Galerkin solver for the 2D high-frequency Helmholtz equation

The method of polarized traces provides the first documented algorithm with truly scalable complexity for the highfrequency Helmholtz equation, i.e., with a runtime sublinear in the number of volume unknowns in a parallel environment. However, previous versions of this method were either restricted to a low order of accuracy, or suffered from computationally unfavorable boundary reduction to ρ(p) interfaces in the p-th order case. In this note we rectify this issue by proposing a high-order method of polarized traces with compact reduction to two, rather than ρ(p), interfaces. This method is based on a primal Hybridizable Discontinuous Galerkin (HDG) discretization in a domain decomposition setting. 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 discontinuous coefficients, and it allows for adaptive refinement in h and p. High order of accuracy is very important for attenuation of the pollution error, even in settings when the medium is not smooth. We provide some examples to corroborate the convergence and complexity claims. Keywords: finite element; frequency-domain; numerical; acoustic; wave equatio

SlabLU: A Two-Level Sparse Direct Solver for Elliptic PDEs

The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two dimensional domain. The solver is designed to reduce communication costs and perform well on GPUs; it uses a two-level framework, which is easier to implement and optimize than traditional multi-frontal schemes based on hierarchical nested dissection orderings. The scheme decomposes the domain into thin subdomains, or "slabs". Within each slab, a local factorization is executed that exploits the geometry of the local domain. A global factorization is then obtained through the LU factorization of a block-tridiagonal reduced coefficient matrix. The solver has complexity $O(N^{5/3})$ for the factorization step, and $O(N^{7/6})$ for each solve once the factorization is completed. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order convergent multi-domain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size $1000 \lambda \times 1000 \lambda$ (for which N=100 \mbox{M}) is solved in 15 minutes to 6 correct digits on a high-powered desktop with GPU acceleration

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched