An adaptive finite element multigrid solver using GPU acceleration

TL;DR

This paper presents a GPU-accelerated adaptive finite element multigrid solver that significantly speeds up complex PDE simulations, achieving up to 20 times faster performance than CPU-based methods.

Contribution

It introduces a method to efficiently integrate GPU acceleration into existing finite element software using linear algebra, with minimal code changes.

Findings

Achieved up to 20X speedup over CPU implementations.

Successfully applied to transport-diffusion, elasticity, and Navier-Stokes equations.

Demonstrated effective GPU utilization with minimal code modifications.

Abstract

Adaptive finite elements combined with geometric multigrid solvers are one of the most efficient numerical methods for problems such as the instationary Navier-Stokes equations. Yet despite their efficiency, computations remain expensive and the simulation of, for example, complex flow problems can take many hours or days. GPUs provide an interesting avenue to speed up the calculations due to their very large theoretical performance. However, the large degree of parallelism and non-standard API make the use of GPUs in scientific computing challenging. In this work, we develop a GPU acceleration for the adaptive finite element library Gascoigne and study its effectiveness for different systems of partial differential equations. Our goal is thereby to integrate the GPU acceleration into the existing code with minimal changes, even when this requires a penalty in the GPU acceleration. Through the systematic formulation of all computations as linear algebra operations, we can employ GPU-accelerated linear algebra libraries, which simplifies the implementation and ensures the maintainability of the code while achieving very efficient GPU utilizations. Our results for a transport-diffusion equation, linear elasticity, and the instationary Navier-Stokes equations show substantial speedups of up to 20X compared to multi-core CPU implementations.