Structured-grid multigrid with Taylor-Hood finite elements

Abstract

Recent years have seen renewed interest in the numerical solution of the Stokes Equations. At the same time, new computational architectures, such as GPUs and manycore processors, naturally perform best with the regular data access and computation patterns associated with structured-grid discretisations and algorithms. While many preconditioning approaches ignore the underlying mesh geometry, our approach is to develop a structured-grid implementation, taking advantage of the highly structured data-access patterns and employing stencil-based calculations. This opens up many opportunities for fine-grained parallelism, allowing us to take advantage of multicore and accelerated architectures. In this thesis, we will consider an implementation of a structured-grid monolithic Multigrid approach for Q2-Q1 finite-element discretisations, comparing its efficiency to an unstructured grid solver implemented in Trilinos. With the aim to eventually target large heterogeneous systems, we will discuss an implementation for moving from a serial code to the GPU by means of OpenCL and compare the efficiency of all three versions. Speedup factors of about 6.3x were observed for the GPU implementation over a serial implementation in Trilinos for a problem on a 768x768 mesh in 2D.