A Volume Integral Equation Stokes Solver for Problems with Variable Coefficients

Abstract

We present a novel numerical scheme for solving the Stokes equation with variable coefficients in the unit box. Our scheme is based on a volume integral equation formulation. Compared to finite element methods, our formulation decouples the velocity and pressure, generates velocity fields that are by construction divergence free to high accuracy and its performance does not depend on the order of the basis used for discretization. In addition, we employ a novel adaptive fast multipole method for volume integrals to obtain a scheme that is algorithmically optimal. Our scheme supports non-uniform discretizations and is spectrally accurate. To increase per node performance, we have integrated our code with both NVIDIA and Intel accelerators. In our largest scalability test, we solved a problem with 20 billion unknowns, using a 14-order approximation for the velocity, on 2048 nodes of the Stampede system at the Texas Advanced Computing Center. We achieved 0.656 petaFLOPS for the overall code (23% efficiency) and one petaFLOPS for the volume integrals (33% efficiency). As an application example, we simulate Stokes flow in a porous medium with highly complex pore structure using a penalty formulation to enforce the no slip condition.

PVFMM: A Parallel Kernel Independent FMM for Particle and Volume Potentials

“…Our method can also be used to solve variable coefficient elliptic PDEs by solving the integral formulation using iterative linear solvers. The details of that work can be found in [20].…”

An efficient preconditioner for the fast simulation of a 2D stokes flow in porous media

“…Other groups have used integral equations to simulate porous media by using a Darcy approximation, 26,27 or by coupling a penalization method with a volume integral equation. 28 In contrast, we will use a BIE to solve the incompressible Stokes equations.…”

“…Furthermore, a natural extension of this work is to solve the three‐dimensional Stokes equations while only discretizing the two‐dimensional boundary of the geometry. Other groups have used integral equations to simulate porous media by using a Darcy approximation, () or by coupling a penalization method with a volume integral equation . In contrast, we will use a BIE to solve the incompressible Stokes equations.…”

Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation

“…Application areas of FMM for discrete volume integrals are astrophysics [14], Brownian dynamics [73], classical molecular dynamics [82], density functional theory [88], vortex dynamics [104], and force directed graph layout [105]. FMM for continuous volume integrals have been used to solve Schrödinger [106] and Stokes [77] equations. More generalized forms of FMM can be used as fast kernel summation for Bayesian inversion [3], Kalman filtering [72], Machine learning [47,70], and radial basis function interpolation [52].…”