Mathematical Sciences: Advanced Spectral Formulations for the Boundary Integral Method

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9312308 Higdon The investigator develops robust spectral boundary integral algorithms for three-dimensional transport problems in realistic geometries. Thw metjhods combine the high-order convergence of spectral methods with the geometric adaptability of boundary element methods. Suitability for parallel computers is a central aspect of the project. The methods are applied to the study of effective transport propertiues of fibrous porous media. Boundary integral methods convert partial differential equations, which describe a phenomenon of interest in a region, into integral equations whose solutions are determined by certain functions on the boundary of the region. Spectral methods break up the boundary into smaller pieces, on which the functions are approximated by polynomials of high degree. Higher-degree polynomials approximate the functions more accurately. The use of polynomials allows for fast calculation of an approximate solution. For problems involving regions of complicated geometry, combining these ideas produces a computational method of significant flexibility and accuracy. The method is used to calculate properties of fluid flow and transport in fibrous porous media. Such problems arise throughout the chemical industry and in many fundamental biological contexts -- for example, in the pulp and paper industry, and in the study of fluid circulation in plants.