A new spectral method approach for singular integral equations

该研究计划的重点是介绍一种新的，快速的，光谱精确的算法，用于解决复杂的一维边界上的一般奇异积分方程，它允许在两个空间维度的椭圆偏微分方程的解的表示。奇异积分方程在电磁学和地震成像的声散射、断裂力学、流体动力学和束物理学中有着丰富的历史。 该方案的成果将在Julia编程语言编写的开源软件包中实施。边界密度的端点奇异性的理论确定允许直接求解器在不使用H-P自适应细化的情况下获得谱精确的全局解。通过成功地进一步开发一类新的直接求解器，该软件包将以稳定和及时的方式解决各种奇异积分方程。 最近推出的直接求解器将结合层次求解器的基础上递归块对角化通过舒尔补。这将特别利用分层非对角低秩结构所产生的强制奇异积分算子的椭圆型偏微分方程。分层求解器涉及独立于强制项的预计算阶段。一旦这种预计算将算子分解，许多强制项的求解具有较低的复杂性，因此需要一小部分时间。 这个程序也将考虑定义在一类重要边界上的奇异积分方程：那些是单位区间和圆的多项式映射。将进行相当多的分析，再次获得带状奇异积分算子通过谱映射定理。求解混合边界条件或多连通边界条件下的奇异积分方程，会导致在交界处具有复杂代数奇异结构的分段定义解。这些困难将接近设计基地，充分捕捉这种复杂的奇异结构出现在交界处。 新的谱方法将用于求解Stokes流、双调和方程和断裂力学的应力应变计算问题。新的谱方法与稳定的高阶时间步进计划相结合，将允许探索的Helmholtz方程的时域积分公式和Rayleigh-Taylor不稳定性的模拟。它还将允许实验和潜在的发现新的现象，在重要的应用，如在纳米级的光学元笼，逆散射问题的解决方案，并在深水内波的BenjaminOno方程的模拟。

This research programme is centred around the introduction of a new, fast, and spectrally accurate algorithm for solving general singular integral equations on complicated one-dimensional boundaries, which allows for a representation of the solution of elliptic partial differential equations in two spatial dimensions. Singular integral equations have a rich history in acoustic scattering for electromagnetics and seismic imaging, fracture mechanics, fluid dynamics, and beam physics. Results of the programme will be implemented in an open source package written in the Julia programming language. Theoretical determination of the endpoint singularities of the boundary densities allows for the direct solver to obtain spectrally accurate global solutions without the use of h-p adaptive refinement. By successfully furthering the development of a new class of direct solvers, the software package will solve a wide range of singular integral equations in a stable and timely manner. The recently introduced direct solver will be combined with a hierarchical solver based on recursive block diagonalization via Schur complements. This will specifically exploit the hierarchically off-diagonal low-rank structure arising from coercive singular integral operators of elliptic partial differential equations. The hierarchical solver involves a pre-computation phase independent of the forcing term. Once this pre-computation factorizes the operator, the solution to many forcing terms has a lower complexity and therefore takes a fraction of the time. This programme will also consider singular integral equations defined on an important class of boundaries: those that are polynomial maps from the unit interval and circle. A considerable analysis will be performed to again obtain banded singular integral operators via the spectral mapping theorem. Solving singular integral equations with either mixed boundary conditions or multiply connected contours leads to piecewise-defined solutions with complicated algebraic singular structure at the junctions. These difficulties will be approached by designing bases that fully capture this complicated singular structure arising at the junctions. The new spectral method will be applied to solve problems of Stokes flow, the biharmonic equation and stress and strain computations for fracture mechanics. Combination of the new spectral method with stable and high-order time-stepping schemes will allow for the exploration of time-domain integral formulations of the Helmholtz equation and the simulation of RayleighTaylor instability. It will also allow experimentation and potential discovery of new phenomena in important applications such as optical metacages at the nanoscale, the solution of inverse scattering problems, and simulation of the BenjaminOno equation for internal waves in deep water.