Integral Equation Methods for Variable Coefficient Elliptic Problems and Applications

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

The focus of this proposal is on the mathematical analysis and efficient implementation of integral equation based methods (IEM) for the solution of physically important, spatially inhomogeneous or "variable coefficient" elliptic equations. These arise in applications which require the modeling of complex fluid flows, such as combustion or ground water pollution. They also arise in the design of novel fluidic Micro-Electro-Mechanical Systems (MEMS), as well as solid-state semiconductor devices. Over the last decade, IEMs have proven to be very successful and are now widely used in electromagnetic, elastic, and fluid dynamic modeling when the governing equation is of "constant coefficient" type. In order to address more general, variable coefficient problems, we represent the solution as a volume integral using the Green's function for a simple, nearby problem convolved with an unknown density. Inserting this representation into the original elliptic equation leads to a linear integral equation. A principal advantage of this approach is that it avoids solving the sparse but poorly conditioned linear systems that result from direct discretization of the differential equation. Discretization of the linear integral equation leads, instead, to a well conditioned dense linear system, for which Krylov subspace methods converge quickly. Historically, the disadvantage of using iterative methods with integral equations was the cost of computing the dense matrix-vector multiplications which are needed at every step. With the advent of the fast multipole method (FMM), however, these can be evaluated in optimal time. Nevertheless, in order for the method to be robust and practical, several issues remain. In particular, effective preconditioning strategies for problems with strong gradients in the material properties are not well developed. Preliminary analysis of some multilevel approaches for model problems are promising. Our planned research program combines classical mathematics (PDEs, integral equations, potential theory) with scientific computing research (fast summation, iterative solvers, adaptive mesh refinement, domain decomposition) and important applications (porous media and variable density flows). The interdisciplinary nature of the work provides fertile ground for training graduate students as computational scientists. Experience gained in the application areas mentioned will have broad impact, since the techniques created can be transferred to many other application areas in the environmental, geophysical, biological, and engineering sciences.