Fast Semi-Iterative Finite Element Poisson Solvers for Tensor Core GPUs Based on Prehandling

The impetus for the research presented in this work is provided by recent developments in the field of GPU computing. Nvidia GPUs that are equipped with Tensor Cores, such as the A100 or the latest H100, promise an immense computing power of 156 and 495 TFLOPS, respectively, but only for dense matrix operations carried out in single precision (with even higher rates in half precision), since this serves their actual purpose of accelerating AI training. It is shown that this performance can also be exploited to a large extent in the domain of matrix-based finite element methods for solving PDEs, if specially tailored, hardware-oriented methods are used. Such methods need to preserve sufficient accuracy, even if single precision is used, and mostly consist of dense matrix operations. A semi-iterative method for solving Poisson’s equation in 2D and 3D based on prehandling, i.e., explicit preconditioning, by means of hierarchical finite elements or generating systems, that satisfies these requirements, is derived and analyzed.Actual benchmark results on an H100 allow the determination of optimal solver configurations in terms of performance, which ultimately exceeds that of a standard geometric multigrid solver on CPU

An extension of a very fast direct finite element Poisson solver on lower precision accelerator hardware towards semi-structured grids

Graphics cards that are equipped with Tensor Core units designed for AI applica tions, for example the NVIDIA Ampere A100, promise very high peak rates concerning their computing power (156 TFLOP/s in single and 312 TFLOP/s in half precision in the case of the A100). This is only achieved when performing arithmetically intensive operations such as dense matrix multiplications in the aforementioned lower precision, which is an obstacle when trying to use this hardware for solving linear systems arising from PDEs discretized with the finite element method. In previous works, we delivered a proof of concept that the predecessor of the A100, the V100 and its Tensor Cores, can be exploited to a great extent when solving Poisson’s equation on the unit square if a hardware-oriented direct solver based on prehandling via hierarchical finite elements and a Schur complement approach is used. In this work, using numerical results on an A100 graphics card, we show that the method also achieves a very high performance if Poisson’s equation, which is discretized by linear finite elements, is solved on a more complex domain corresponding to a flow around a square configuration