GPU-accelerated Linear Solvers for High-order Finite Element Methods in Poisson Problems

Abstract

Solving large-scale linear systems arising from high-order finite element discretizations for Poisson equations often represents the most expensive component of the high-order finite element solver. This dissertation develops and analyzes numerical methods and algorithms aimed at improving the convergence and efficiency of the preconditioned conjugate gradient (PCG) algorithm for high-order finite element methods on Graphics Processing Units (GPUs).

The conjugate gradient algorithm iteratively refines an initial approximate solution until a specified stopping criterion is met, with its convergence rate enhanced through the application of a preconditioner. Novel smoothers are constructed within a multigrid preconditioner to improve the convergence rate on highly deformed meshes. Additionally, new stopping criteria are introduced to balance various error sources, thereby reducing the number of iterations. An adaptive mixed precision conjugate gradient algorithm is proposed to exploit the superior computational performance of GPUs at lower precisions while maintaining convergence and accuracy. Furthermore, a comprehensive optimization of the PCG algorithm, including a careful examination of the memory hierarchy, is developed. The effectiveness of these novel approaches is demonstrated for GPU-accelerated high-order finite element discretizations in Poisson problems.