Mathematical Sciences: Advanced Spectral Formulations for the Boundary Integral Method

小行星9312308 研究人员开发了强大的光谱边界积分算法的三维运输问题在现实的几何形状。 该方法将谱方法的高阶收敛性与边界元方法的几何适应性结合起来，具有联合收割机的优点。 并行计算机的适用性是该项目的一个核心方面。 并将其应用于纤维多孔介质有效输运特性的研究。 边界积分方法将描述区域中感兴趣现象的偏微分方程转换为积分方程，积分方程的解由区域边界上的某些函数确定。 谱方法将边界分解成更小的块，在这些块上，函数由高次多项式近似。 高次多项式更精确地逼近函数。 多项式的使用允许近似解的快速计算。 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。 用该方法计算了纤维多孔介质中流体的流动和输运特性。 这样的问题出现在整个化学工业和许多基本的生物学背景下-例如，在纸浆和造纸工业，并在植物中的流体循环的研究。

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.