A Volume Integral Equation Stokes Solver for Problems with Variable Coefficients

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 peta FLOPS for the overall code (23% efficiency) and one peta FLOPS for the volume integrals (33% efficiency). As an application example, we simulate Stokes ow in a porous medium with highly complex pore structure using a penalty formulation to enforce the no slip condition.