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 peak 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. 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.

翻译：自适应有限元与几何多重网格求解器的结合是处理非稳态Navier-Stokes方程等问题最高效的数值方法之一。然而，尽管计算效率较高，其计算成本依然高昂，例如复杂流动问题的模拟可能需要数小时甚至数天。GPU因其极高的理论峰值性能，为加速计算提供了极具前景的途径。然而，GPU的高度并行性及非标准应用程序接口（API）为其在科学计算中的应用带来了挑战。本研究针对自适应有限元库Gascoigne开发了GPU加速方案，并研究了其在不同偏微分方程系统中的有效性。通过将所有计算系统地转化为线性代数运算，我们得以采用GPU加速的线性代数库，这不仅简化了实现过程，确保了代码的可维护性，同时实现了极高的GPU利用率。针对输运-扩散方程、线弹性力学及非稳态Navier-Stokes方程的测试结果表明，与多核CPU实现相比，本方法实现了高达20倍的显著加速效果。