A matrix-free macro-element variant of the hybridized discontinuous Galerkin method

TL;DR

This paper introduces a macro-element variant of the hybridized discontinuous Galerkin method that enhances scalability and parallelism for solving PDEs, especially suited for high-performance computing environments.

Contribution

It develops a macro-element HDG method allowing local refinement, parallel local problems, and matrix-free global solves, improving scalability and efficiency.

Findings

Efficient parallelization on n-node clusters.

Reduced global degrees of freedom with larger local problems.

Effective matrix-free iterative solution for the global problem.

Abstract

We investigate a macro-element variant of the hybridized discontinuous Galerkin (HDG) method, using patches of standard simplicial elements that can have non-matching interfaces. Coupled via the HDG technique, our method enables local refinement by uniform simplicial subdivision of each macro-element. By enforcing one spatial discretization for all macro-elements, we arrive at local problems per macro-element that are embarrassingly parallel, yet well balanced. Therefore, our macro-element variant scales efficiently to n-node clusters and can be tailored to available hardware by adjusting the local problem size to the capacity of a single node, while still using moderate polynomial orders such as quadratics or cubics. Increasing the local problem size means simultaneously decreasing, in relative terms, the global problem size, hence effectively limiting the proliferation of degrees of freedom. The global problem is solved via a matrix-free iterative technique that also heavily relies on macro-element local operations. We investigate and discuss the advantages and limitations of the macro-element HDG method via an advection-diffusion model problem.