Efficient and highly accurate solvers for integral equations on surfaces with edges and corners

University Of California-Davis, Davis CA

Investigators

Abstract

This project concerns the computer modeling of physical phenomena. In particular, the PI seeks to accurately model the behavior of electromagnetic and acoustic waves. This topic has a long history, and many important results have been previously achieved, but we currently lack the ability to accurately model situations involving complex geometry. This project seeks to address these difficulties by combing several observations from pure mathematics with new engineering approaches. The end result of this project will be tools which allow for the accurate modeling of electromagnetic and acoustic waves. These tools will be applicable to many problems, but the PI is particularly interested in applying them to integrated circuit analysis and biomechanical simulations (for instance, vesicle flows). Many of the partial differential equations of mathematical physics can be profitably reformulated as integral equations. Such methods have important applications to problems in electrodynamics, fluid dynamics and elasticity. However, most applications involve domains with singularity, and it is notoriously difficult to achieve high-accuracy and efficiency when solving integral equations on such domains. In principle, it appears that there should be no difficulty in solving a large class of integral equations given on surfaces with edge and corner singularities in a brute-force fashion. Many important boundary value problems in mathematical physics can be formulated using integral operators which are invertible and well-conditioned on spaces of square integral functions. Galerkin discretizations of such operators converge and are as well-conditioned as the underlying operator. It follows from these observations that, assuming all aspects of discretization are correctly handled, simply representing solutions locally with polynomial basis functions on a sufficiently dense mesh will result in highly accurate approximations. However, several difficult problems arise in practice with this brute-force approach; chief among them are: (1) dense meshes lead to excessively large linear systems that even modern O(N) fast solvers are inadequate to address; (2) representing singularities near edges is best done with highly anisotropic meshes which cause difficulties for currently available discretization techniques and fast solvers; (3) evaluating the entries of coefficient matrices to high accuracy, which involves estimating singular and "nearly" singular integrals, is quite challenging in general and substantially more so near corner and edge regions. The goal of this project is to develop highly-accurate fast robust solvers for integral equations on surfaces with edge and corner singularities which overcome these difficulties and achieve high-accuracy and efficiency. It will do so without the use of a priori asymptotic estimates (which are not available in many cases of interest). The project consists of a three-phased approach: (1) The PI will implement a highly-accurate and very robust "brute-force" procedure; (2) several tools, including local fast adaptive mesh generators and operator compression techniques will be deployed in order to accelerate the brute-force solver; (3) finally, efficient numerically precomputed quadrature formulae, which characterize the singularities of solutions and serve as a substitutes for a priori asymptotic estimates, will be constructed using the accelerated solver.