Finite Element Integration with Quadrature on the GPU

TL;DR

This paper introduces a GPU-optimized quadrature-based finite element integration method that achieves high computational throughput for low-order elements, significantly improving efficiency in PDE solvers.

Contribution

The authors develop a novel thread transposition pattern for finite element integration on GPUs, enabling high vectorization and avoiding reductions, with demonstrated near-peak performance.

Findings

Achieves 300 GF/s in 2D and 400 GF/s in 3D for Laplacian integration

Performance closely matches the bandwidth-limited model predictions

Effective for vector-valued PDEs like linear elasticity

Abstract

We present a novel, quadrature-based finite element integration method for low-order elements on GPUs, using a pattern we call \textit{thread transposition} to avoid reductions while vectorizing aggressively. On the NVIDIA GTX580, which has a nominal single precision peak flop rate of 1.5 TF/s and a memory bandwidth of 192 GB/s, we achieve close to 300 GF/s for element integration on first-order discretization of the Laplacian operator with variable coefficients in two dimensions, and over 400 GF/s in three dimensions. From our performance model we find that this corresponds to 90\% of our measured achievable bandwidth peak of 310 GF/s. Further experimental results also match the predicted performance when used with double precision (120 GF/s in two dimensions, 150 GF/s in three dimensions). Results obtained for the linear elasticity equations (220 GF/s and 70 GF/s in two dimensions, 180 GF/s and 60 GF/s in three dimensions) also demonstrate the applicability of our method to vector-valued partial differential equations.