Multiscale Methods for the Rapid Solution of Boundary Integral Equations in Geometrically Complicated Domains

Southern Methodist University, Dallas TX

Investigators

Abstract

The proposed research is concerned with multiscale discretizations of boundary integral operators on geometrically complicated surfaces. The standard approach, generating multiscale discretizations in parameter spaces and lifting them on the surface, is not efficient when a large number of parameter patches is required. This project is concerned with an alternative approach to multiscale discretizations. The basis is generated in a hierarchical decomposition of the three-space and subsequently restricted on the boundary surface. This construction leads to a sparse representation of the integral operator even for complicated geometries. It is planned to apply this approach to boundary integral formulations of general elliptic as well as time dependent parabolic problems. The range of important engineering applications which the proposed research could impact is quite diverse. Examples include the analysis of integrated circuit interconnect and micromechanical systems. In these areas finding computationally efficient numerical methods for complex three dimensional structures is important in order to generate prototypes interactively on a computer. In the past the major computational tool has been the Fast Multipole Method (FMM), because of its relative simplicity and its flexibility to geometry and integral operator. We construct a multiscale basis by using a hierarchical decomposition of the three-space. The same hierarchy of cubes is also used by the FMM, and therefore the proposed approach is able to combine the flexibility of the FMM with the strengths of multiscale methods. Thus the planned research can lead to faster and more robust algorithms as well as to a better understanding of theoretical and practical issues of both approaches.