Parallel implementation of a compatible high-order meshless method for the Stokes' equations

TL;DR

This paper presents a scalable, high-order meshless method for steady-state Stokes problems, leveraging GPU acceleration and parallel computing to achieve efficient, accurate solutions with favorable convergence and scalability.

Contribution

The work introduces a parallel, high-order meshless discretization scheme for Stokes equations that uses local matrix inversions and GPU acceleration, improving scalability and convergence.

Findings

Achieves high-order convergence for velocity and pressure.

Demonstrates high scalability and efficiency on parallel architectures.

Uses local matrix inversion and GPU acceleration for stencil generation.

Abstract

A parallel implementation of a compatible discretization scheme for steady-state Stokes problems is presented in this work. The scheme uses generalized moving least squares to generate differential operators and apply boundary conditions. This meshless scheme allows a high-order convergence for both the velocity and pressure, while also incorporates finite-difference-like sparse discretization. Additionally, the method is inherently scalable: the stencil generation process requires local inversion of matrices amenable to GPU acceleration, and the divergence-free treatment of velocity replaces the traditional saddle point structure of the global system with elliptic diagonal blocks amenable to algebraic multigrid. The implementation in this work uses a variety of Trilinos packages to exploit this local and global parallelism, and benchmarks demonstrating high-order convergence and weak scalability are provided.