PYI: Computational Issues in the Solution of Partial Differential Equations

偏微分方程的数值求解是目前计算难度最大的任务之一，因此只能采用最有效的算法。目前，算法中实际使用的数值公式已被很好地理解，但由几何问题引起的计算问题尚不清楚。一项研究旨在确定如何正确处理偏微分方程的网格。求解技术，如有限元方法，需要将几何域细分为小的多边形单元。本研究将为实现这一细分提供最佳的方法，特别是当区域具有复杂的形状时。它将确定在并行计算机的处理器之间划分域的适当方法。边界元技术在处理复杂几何图形方面显示出巨大的前景。由于新发现了它们与小波变换的联系，它们最近受到了关注。这项研究寻求能够加快边界元素计算速度的新算法。

Numerical solution of partial differential equations is one of the most computationally difficult tasks in existence, so only the most efficient algorithms can be used. The actual numerical formulas used in the algorithms are fairly well understood by now, but the computational issues arising from geometrical issues are less so. One line of research is intended to determine the proper handling of grids for partial differential equations. Solution techniques such as finite element methods require the subdivision of the geometric domain into small polygonal elements. This research will establish the best ways to carry out this subdivision, especially when the domain has a complicated shape. It will determine the proper method to partition the domain among processors of a parallel computer. Boundary element techniques show great promise for handling complicated geometries. They are receiving recent attention because of their newly-discovered links to wavelet transformations. The research seeks new algorithms that will speed up boundary element calculations.