This project involves a novel formulation for constructing boundary integral methods to solve boundary value problems involving linear partial differential equations on domains with oriented, piecewise smooth boundaries defined implicitly by the corresponding signed distance functions. The proposed framework will facilitate computations in a wide class of computational problems that involve, e.g., the solutions of Poisson's equation or Helmholtz equation defined on time dependent domains with irregular boundaries. Such computational problems are found in multiphase viscous fluid flows, inverse scattering, shape optimization problems, and gradient flows of surface energies modeling e.g. solidification process of a fluid. The aim of our new formulation is to lift the overhead of remeshing that is needed in finite element methods, and to avoid delicate and possibly complicated formulas found in finite difference based methods that depend on how surfaces "cut" through the underlying grids. The formulation is based on averaging a one parameter family of parameterizations of an integral equation defined on the boundary of the domain. By application of the coarea formula, a novel boundary integral equation without any explicit parameterization of the boundaries is derived. The resulting numerical algorithm is simple and is applicable to a variety of meshing or grids. The proposed research program consists of a systematic study of such new type of boundary integral formulation, encompassing (a) numerical integration methods (quadratures) for singular integrals, (b) analytical and numerical treatment of corners and higher co-dimensional manifolds, (c) applications to nonlinear interface dynamics and shape optimizations.
The proposed research will contribute directly to a crucial mathematical and computational part of many important applications in science and engineering, from multiphase fluids, seismic imaging in petroleum engineering, inverse scattering in wave propagation, high order nonlinear interface evolution found in the study of solidification of fluids, to bio-mechanical applications. The training of students and post-doctoral researchers provided by the proposed research program will allow them to conduct research in highly inter-disciplinary projects and bring state-of-the-art numerical analysis and computational algorithms to the related areas.
