GPU-accelerated discontinuous Galerkin methods on polytopic meshes

TL;DR

This paper introduces GPU-accelerated algorithms for efficiently assembling matrices in discontinuous Galerkin methods on complex polytopic meshes, enabling scalable and fast computations for various PDE problems.

Contribution

It presents novel parallel assembly algorithms on CUDA GPUs for polytopic dG methods, significantly improving computational efficiency and scalability.

Findings

Near-linear scalability with GPU cores for quadrature step

Efficient implementation reduces assembly overhead compared to standard meshes

Applicable to various linear PDE problems with polytopic meshes

Abstract

Discontinuous Galerkin (dG) methods on meshes consisting of polygonal/polyhedral (henceforth, collectively termed as \emph{polytopic}) elements have received considerable attention in recent years. Due to the physical frame basis functions used typically and the quadrature challenges involved, the matrix-assembly step for these methods is often computationally cumbersome. To address this important practical issue, this work proposes two parallel assembly implementation algorithms on CUDA-enabled graphics cards for the interior penalty dG method on polytopic meshes for various classes of linear PDE problems. We are concerned with both single GPU parallelization, as well as with implementation on distributed GPU nodes. The results included showcase almost linear scalability of the quadrature step with respect to the number of GPU-cores used since no communication is needed for the assembly step. In turn, this can justify the claim that polytopic dG methods can be implemented extremely efficiently, as any assembly computing time overhead compared to finite elements on `standard' simplicial or box-type meshes can be effectively circumvented by the proposed algorithms.