Multilevel Interior Penalty Methods on GPUs

TL;DR

This paper introduces a GPU-accelerated multigrid method for high-order DG finite element methods, optimizing data layouts and smoothing techniques, achieving high arithmetic throughput, and demonstrating efficiency in solving Poisson problems.

Contribution

It presents a novel matrix-free multigrid method optimized for GPUs, including performance analysis, mixed-precision strategies, and parallelization for high-order DG methods.

Findings

Achieved up to 39% of GPU peak arithmetic throughput.

Validated effectiveness of mixed-precision and MPI parallelization.

Demonstrated solver robustness in 2D and 3D Poisson problems.

Abstract

We present a matrix-free multigrid method for high-order discontinuous Galerkin (DG) finite element methods with GPU acceleration. A performance analysis is conducted, comparing various data and compute layouts. Smoother implementations are optimized through localization and fast diagonalization techniques. Leveraging conflict-free access patterns in shared memory, arithmetic throughput of up to 39% of the peak performance on Nvidia A100 GPUs are achieved. Experimental results affirm the effectiveness of mixed-precision approaches and MPI parallelization in accelerating algorithms. Furthermore, an assessment of solver efficiency and robustness is provided across both two and three dimensions, with applications to Poisson problems.