Fast integral equation solver for variable coefficient elliptic PDEs in complex geometries

Abstract

This dissertation presents new numerical algorithms and related software for the numerical solution of elliptic boundary value problems with variable coefficients on certain classes of geometries. The target applications are problems in electrostatics, fluid mechanics, low-frequency electromagnetic and acoustic scattering. We present discretizations based on integral equation formulations which are founded in potential theory and Green's functions. Advantages of our methods include high-order discretization, optimal algorithmic complexity, mesh-independent convergence rate, high-performance and parallel scalability.

First, we present a parallel software framework based on kernel independent fast multipole method (FMM) for computing particle and volume potentials in 3D. Our software is applicable to a wide range of elliptic problems such as Poisson, Stokes and low-frequency Helmholtz. It includes new parallel algorithms and performance optimizations which make our volume FMM one of the fastest constant-coefficient elliptic PDE solver on cubic domains. We show that our method is orders of magnitude faster than other N-body codes and PDE solvers. We have scaled our method to half-trillion unknowns on 229K CPU cores.

Second, we develop a high-order, adaptive and scalable solver for volume integral equation (VIE) formulations of variable coefficient elliptic PDEs on cubic domains. We use our volume FMM to compute integrals and use GMRES to solve the discretized linear system. We apply our method to compute incompressible Stokes flow in porous media geometries using a penalty function to enforce no-slip boundary conditions on the solid walls. In our largest run, we achieved 0.66 PFLOP/s on 2K compute nodes of the Stampede system (TACC).

Third, we develop novel VIE formulations for problems on geometries that can be smoothly mapped to a cube. We convert problems on non-regular geometries to variable coefficient problems on cubic domains which are then solved efficiently using our volume FMM and GMRES. We show that our solver converges quickly even for highly irregular geometries and that the convergence rates are independent of mesh refinement.

Fourth, we present a parallel boundary integral equation solver for simulating the flow of concentrated vesicle suspensions in 3D. Such simulations provide useful insights on the dynamics of blood flow and other complex fluids. We present new algorithmic improvements and performance optimizations which allow us to efficiently simulate highly concentrated vesicle suspensions in parallel.