Mathematical Sciences: Numerical Analysis and Software for Integral Equations in Three Dimensions

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9403589 Atkinson The focus of the proposal is the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation in three dimensions. The work will involve both the development of further theoretical knowledge on the numerical methods being used and the further development of a large computer package (BIEPACK) to solve BIE on surfaces in three dimensions. The numerical analysis problems to be studied include the further development of theory for the approximation of surfaces, the numerical integration of singular and nonsingular functions over surfaces, the use of graded meshes to approximate functions that are ill-behaved around edges and corners of the surface, and the iterative solution of the large dense linear systems of equations which arise when discretizing the BIE. In addition, new topics involving the use and numerical solution of BIE are to be considered. These include: (1) the use of BIE of the first kind in solving the Dirichlet and Neumann problems for Laplace's equation; (2) the approximation and solution of hypersingular BIE; and (3) the use of "clustering methods" to reduce the cost of the numerical integration and matrix-vector multiplications that are a part of most numerical methods. Many of the results of these studies will be incorporated into the present package BIEPACK, to both improve its present programs and to extend it to new BIE. It is also planned to apply numerical methods that were developed for BIE to the solution of integral equations arising in computer graphics, and in particular, the "rendering equation". Many engineering problems involve the solution of a mathematical equation called "Laplace's equation" or perturbations of it; and the name "potential theory" is given to the study and application of Laplace's equation and closely related equations. The Laplace equation itself occurs in applications such as fluid mechanics (irrotational incompressible fluid flows), heat transfer, gravit ational fields, electrostatic fields, and others. Variations on Laplace's equation occur in fluid mechanics (Stokes' flows), linear elasticity theory, electromagnetic and acoustic wave propagation, and other topics. The solution of Laplace's equation is a function defined for all points of some region in space, and the objective is to find this function. In three dimensions, this can be a very time-consuming process. Many analytical and numerical techniques are used to solve Laplace's equation and related equations, including finite difference methods, finite element methods, and boundary integral equation methods. One of the advantages of the latter is that the problem is re-defined as a new equation over the boundary of the original region on which the Laplace equation of interest is defined, and this can often lead to time savings in the overall solution process. The new problem is called a "boundary integral equation" or BIE. The intention of the present study is to improve numerical methods for the numerical solution of BIE, as well as to develop a further theoretical understanding of numerical methods for solving BIE. These results will be used to improve a package, called BIEPACK, that has been developed for the numerical solution of many BIE for Laplace's equation. It is also planned to apply numerical methods that were developed for BIE to the solution of integral equations arising in computer graphics.