Mathematical Sciences: Advanced Spectral Formulations for the Boundary Integral Method

9312308 Higdon研究者为现实几何中的三维输运问题开发了鲁棒的谱边界积分算法。该方法将谱法的高阶收敛性与边界元法的几何适应性相结合。并行计算机的适用性是该项目的一个中心方面。将该方法应用于纤维状多孔介质的有效输运特性研究。边界积分法将描述区域内感兴趣的现象的偏微分方程转换成解由区域边界上的某些函数决定的积分方程。谱方法将边界分解成更小的块，在这些块上用高阶多项式近似函数。高次多项式更精确地逼近函数。多项式的使用使近似解的快速计算成为可能。对于涉及复杂几何区域的问题，结合这些思想产生了一种具有显著灵活性和准确性的计算方法。该方法用于计算纤维状多孔介质中流体的流动和输运特性。这类问题出现在整个化学工业和许多基本的生物环境中——例如，在纸浆和造纸工业中，以及在植物流体循环的研究中。

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.