解第一类高维边界积分方程的高精度求积法与分裂外推法

Many realistic problems in scientific research and engineering applications come down to either interior or exterior boundary value problems in high dimensions. Based on the single layer potential theory, their mathematical models can be transferred to the first-kind boundary integral equations. This approach can reduce the dimension of the problem, easy to deal with exterior boundary problems, decrease the computational complexity for assembling the discrete coefficient matrices. With the help of certain integral transformations, it can overcome the difficulty caused by strong singularities in problems with the pits or cracks. Furthermore, the accuracy of approximation solutions can be greatly improved by the splitting extrapolation method, while, the efficiency of the method can advance by designing appropriate self-adaptive algorithms. Many numerical experiments have proved that the numerical quadrature method for solving the first-kind boundary integral equation outperforms that of the second-kind boundary integral equation in both accuracy and efficiency. However, its mathematical theory is sophisticated: Tao Lu and Jin Huang first developed a rigorous theoretical analysis of the numerical algorithm; Jin Huang et al. proved that the approximation solution to the Laplace、Laplacian、Helmholtz、Stokes equations and so on a polygonal domain can achieve higher accuracy by employing the splitting extrapolation method and an a posteriori error estimator. However, for the first-kind of boundary integral equations related to more complicated problems in three dimensions such as mechanics problems in elasticity, the quadrature methods of high accuracy and splitting extrapolation method is still underdeveloped. Indeed, the solution to high- dimensional boundary integral equations represents one of the the most significant computational hurdles in the large-scale scientific computing. On the other hand, the splitting extrapolation method was first proposed by Chinese scholars and has achieved high evaluations from other researchers in various area all over the world. Therefore, in order to overcome the main computation hurdle, also to maintain our leading position in this active research area, it is worth to provide continuous support for investigating the quadrature methods of high accuracy and splitting extrapolation method in high-dimensional complex problems.

科学与工程中大量问题都归结于解高维内、外边值问题。使用单层位势把问题转换为第一类边界积分方程不仅降低问题的维数、便于处理外问题、降低离散矩阵生成成本、通过积分变换克服凹点和裂缝带来的强奇异困难，而且能够应用分裂外推提高数值解的精度和构造自适应算法。实践证明使用求积法解第一类边界积分方程，无论精度和计算量都优于第二类边界积分方程。但是其理论结构复杂，吕涛和黄晋首先提供算法的理论依据，黄晋等又证明多角形区域Laplace、Helmholtz、Stokes等方程近似解可以使用分裂外推提高精度和后验误差估计。但是对于更复杂弹性力学和三维问题的第一类边界积分方程的求积法与分裂外推算迄今还是空白。鉴于如何解高维边界积分方程是当今大型科学计算难点和核心内容，分裂外推又是中国学者开创且在国际上有很高评价的成果。因此继续支持分裂外推研究向高维和复杂问题纵深发展，以便保持我国在这个领域研究的领先优势，

本项目提供了求解高维弱奇异积分方程（组）的高精度求积法和分裂外推算法，尤其是求解第一类边界积分方程的机械求积法。机械求积法一种新算法，具有许多不可替代的优点。首先，它在克服高维弱奇异性效应时非常有效,不需要计算任何奇异积分，从而减少了计算量，同时能够获得高精度和误差的渐近展开式。其次，这种算法求解积分方程时得到的离散矩阵条件数非常小，是一种特别稳定的算法。此外，它还拥有自适应的后验误差估计。本项目的研究成果主要包括:.1）推导了求解多维奇异和超奇异积分的求积公式及相应的Euler-Maclaurin展开式，并发表2篇SCI期刊论文和2篇EI期刊论文。.2）建立了求解多维弱奇异积分方程（组）的高精度求积法、外推与分裂外推算法理论，并发表3篇SCI期刊论文和2篇EI期刊论文。.3）建立了第一类弱奇异边界积分方程（组）的高精度求积法的基本理论和分裂外推算法理论，并发表7篇SCI期刊论文和3篇EI期刊论文。.4）建立了第一类非线性边界积分方程（组）的高精度求积法、外推与分裂外推算法理论，并发表1篇SCI期刊论文。.5）建立了第二类高维弱奇异Fredholm积分方程（组）的高精度求积法、外推与分裂外推算法理论，并发表4篇SCI期刊论文和1篇EI期刊论文。