The thesis focuses on numerical methods for boundary integral equation (BIE) formulations of partial differential equations (PDEs). The work contains three parts: the first two consider numerical solution methods for boundary integral equations in wave scattering and Stokes flow, respectively. The last part proposes an adaptive discretization technique for BIEs in 2D.
The proposed work is based on previous developments in fast direct solution techniques for BIEs. Such methods exploit the rank deficiency in the off-diagonal blocks of the discretized system and build an approximation to the inverse with linear cost for two-dimensional problems. Once the inverse approximation is constructed, applying it to any given vector is very cheap, making the methods ideal for problems with lots of right-hand-sides. The two direct solvers presented in this thesis are driven by real-life applications. The scattering solver is built to assist practitioners in designing acoustic and optic devices to manipulate waves. Its efficiency in handling multiple incident angles and minor modifications in the structure will be handy in an optimal design setting. The Stokes solver is to help with numerical simulation of objects such as bacteria and vesicles in viscous flow. To accurately capture the interaction between the objects and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the objects. This makes standard fast direct solvers too expensive to be useful, as the linear system changes for each time step. The proposed approach avoids this by pre-constructing a fast direct solver for the wall independently of time and updating the original solver to accommodate any refinements in discretization. The last part of the thesis describes an adaptive discretization technique for two-dimensional BIEs. Standard adaptive discretization method often requires a sequence of global boundary density solves each on a finer grid and terminates with the last grid if the improvements obtained from the next finer level is very small. The global density solves make the cost of the standard approach relatively high. The proposed alternative reduces the cost by replacing global solves with local solves for an approximate of the true density.