A new spectral method approach for singular integral equations

这项研究计划的中心是引入一种新的、快速的、频谱精确的算法来求解复杂一维边界上的一般奇异积分方程组，该算法允许在二维空间中表示椭圆型偏微分方程解。奇异积分方程组在电磁和地震成像、断裂力学、流体力学和波束物理的声散射领域有着丰富的历史。 该方案的结果将在用Julia编程语言编写的开放源码程序包中实施。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光谱上精确的整体解。通过成功地开发一种新的直接求解器，该软件包将以稳定和及时的方式求解各种奇异积分方程组。 最近引入的直接求解器将与基于Schur互补的递归块对角化的分层求解器相结合。这将特别利用椭圆型偏微分方程的强制奇异积分算子所产生的层次化非对角线低阶结构。分层求解器包括一个独立于强迫项的预计算阶段。一旦这个预计算分解了运算符，许多强迫项的解决方案就具有较低的复杂性，因此只需要一小部分时间。 本方案还将考虑定义在一类重要边界上的奇异积分方程组：这些边界是单位区间和圆的多项式映射。为了再次利用谱映射定理得到带状奇异积分算子，将进行大量的分析。求解具有混合边界条件或多连通轮廓线的奇异积分方程组，可得到具有复杂代数奇异结构的分段定义解。这些困难将通过设计完全捕捉在交界处产生的这种复杂的奇异结构的基础来解决。 新的谱方法将被应用于解决断裂力学中的Stokes流、双调和方程和应力应变计算问题。将新的谱方法与稳定和高阶时间步进格式相结合，可以探索Helmholtz方程的时间域积分形式，并模拟RayleighTaylor不稳定性。它还将允许在重要应用中进行实验并可能发现新现象，如纳米尺度的光学亚像、逆散射问题的解决以及深水内波的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.