An Efficient and High-Order Accurate Boundary Integral Solver for the Stokes Equations in Three Dimensional Complex Geometries
Candidate: Lexing Ying
Advisor: Denis Zorin

Abstract

This dissertation presents an efficient and high-order boundary integral solver for the Stokes equations in complex 3D geometries. The targeted applications of this solver are the flow problems in domains involving moving boundaries. In such problems, traditional finite element methods involving 3D unstructured mesh generation expe- rience difficulties. Our solver uses the indirect boundary integral formulation and discretizes the equation using the Nyström method.

Although our solver is designed for the Stokes equations, we show that it can be generalized to other constant coefficient elliptic partial differential equations (PDEs) with non-oscillatory kernels.

First, we present a new geometric representation of the domain boundary. This scheme takes quadrilateral control meshes with arbitrary geometry and topology as input, and produces smooth surfaces approximating the control meshes. Our surfaces are parameterized over several overlapping charts through explicit nonsingular C parameterizations, depend linearly on the control points, have fixed-size local support for basis functions, and have good visual quality.

Second, we describe a kernel independent fast multipole method (FMM) and its parallel implementation. The main feature of our algorithm is that it is based only on kernel evaluation and does not require the multipole expansions of the underlying kernel. We have tested our method on kernels from a wide range of elliptic PDEs. Our numerical results indicate that our method is efficient and accurate. Other ad- vantages include the simplicity of the implementation and its immediate extension to other elliptic PDE kernels. We also present an MPI based parallel implementation which scales well up to thousands of processors.

Third, we present a framework to evaluate the singular integrals in our solver. A singular integral is decomposed into a smooth far field part and a local part that contains the singularity. The smooth part of the integral is integrated using the trape- zoidal rule over overlapping charts, and the singular part is integrated in the polar coordinates which removes or decreases the order of singularity. We also describe a novel algorithm to integrate the nearly singular integrals coming from the evaluation at points close to the boundary.