Manycore Parallel Computing for a Hybridizable Discontinuous Galerkin Nested Multigrid Method

Abstract

We present a heterogeneous computing strategy for a hybridizable discontinuous Galerkin (HDG) geometric multigrid (GMG) solver. Parallel GMG solvers require a combination of coarse grain and fine grain parallelism is utilized to improve time to solution performance. In this work we focus on fine grain parallelism. We use Intel's second generation Xeon Phi (Knights Landing) to enable acceleration. The GMG method achieves ideal convergence rates of 0.2 or less, for high polynomial orders. A matrix free (assembly free) technique is exploited to save considerable memory usage and increase arithmetic intensity. HDG enables static condensation, and due to the discontinuous nature of the discretization, we developed a matrix-vector multiply routine that does not require any costly synchronizations or barriers. Our algorithm is able to attain 80% of peak bandwidth performance for higher order polynomials. This is possible due to the data locality inherent in the HDG method. Very high performance is realized for high order schemes, due to good arithmetic intensity, which declines as the order is reduced.

A high order hybridizable discontinuous Galerkin method for incompressible miscible displacement in heterogeneous media

“…However, iterative coupling and high order time stepping may be utilized to enhance the solution, which increases the number of linear solves that are needed per time step. Efficient linear solvers are required for this class of problems [19], as the dominant cost occurs during this phase of the simulation.…”

A divergence‐conforming hybridized discontinuous Galerkin method for the incompressible Reynolds‐averaged Navier‐Stokes equations

“…Second, we plan to examine both linear and nonlinear system solution strategies with an eye toward continually improving computational efficiency. It should be noted that efficient multilevel solution strategies have been proposed for HDG methods in other application areas . Moreover, a continuous velocity trace may be employed to significantly reduce the total number of DOFs .…”

“…It should be noted that efficient multilevel solution strategies have been proposed for HDG methods in other application areas. 46,47 Moreover, a continuous velocity trace may be employed to significantly reduce the total number of DOFs. 48 Third, we plan to extend the method presented here to other turbulence models, including the k-, k-, and SST models.…”

Comparative Study of Finite Element Methods Using the Time-Accuracy-Size(TAS) Spectrum Analysis

“…Various levels of both h-and p-refinement for the CG and DG methods are considered across different software implementations. We do not consider other viable approaches such as the Hybridizable Discontinuous Galerkin (HDG) method [19,24] or mixed formulations [17,36] but these will be addressed in future work. To this end, let us define E as an element belonging to a mesh E(Ω).…”