Fast and accurate numerical algorithms for boundary value problems of elliptic partial differential equations on open surfaces in three dimensions

The focus of the proposed work is directed toward the development of fast, accurate, and robust computational techniques for solving large-scale scattering problems when the scattering surfaces consist of a collection of open surfaces (i.e., surfaces with boundary). Though such problems are often initially formulated as boundary-value problems of elliptic partial differential equations, integral equations have been employed as one of principal tools for the numerical solution of scattering problems, particularly for exterior problems. Historically, most of the integral equations used have been of the first kind, since numerical instabilities associated with such equations have not been critically important for the relatively small-scale problems that could be handled at the time. The combination of improved hardware with the recent progress in the design of 'fast' algorithms has changed the situation dramatically. Condition numbers of systems of linear algebraic equations resulting from the discretization of integral equations of potential theory have become critical, and the simplest way to limit such condition numbers is by starting with second-kind integral equations. Hence, there is increasing interest in reducing scattering problems to systems of second-kind integral equations on the boundaries of scatterers.

The investigator proposed to apply tools from potential theory and singular integrals to construct second-kind integral-equation (SKIE) formulations for open-surface problems (especially for those whose governing PDEs are the Laplace or Helmholtz equations). After SKIE formulations have been obtained, the investigator plans to apply them to develop and implement efficient and accurate numerical algorithms for open-surface problems, using a combination of iterative solvers and the fast multipole method. Since the Laplace equation and the Helmholtz equation are ubiquitous in applied mathematics and many practical problems involve open surfaces, the proposed research will have broad impacts on many active research fields including acoustic and electromagnetic scattering problems, fluid mechanics, elasticity problems, and inverse-scattering problems. The proposed research is also expected to have long-term impacts on key technologies such as reflecting antennas and integrated circuits.