A new spectral method approach for singular integral equations

奇异积分方程的新谱法

基本信息

批准号：
RGPIN-2017-05514
负责人：
Slevinsky, Richard
金额：
$ 1.02万
依托单位：
University of Manitoba
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2020
资助国家：
加拿大
起止时间：
2020-01-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711678
关键词：
new spectral method approach singular

项目摘要

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.

该研究计划的重点是引入一种新的、快速的、光谱精确的算法，用于求解复杂一维边界上的一般奇异积分方程，该算法允许在两个空间维度上表示椭圆偏微分方程的解。奇异积分方程在电磁学和地震成像、断裂力学、流体动力学和梁物理学的声散射方面有着丰富的历史。该程序的结果将在用 Julia 编程语言编写的开源包中实现。边界密度端点奇点的理论确定允许直接求解器获得光谱精确的全局解，而无需使用 h-p 自适应细化。通过成功地进一步开发新型直接求解器，该软件包将稳定、及时地求解各种奇异积分方程。最近推出的直接求解器将与基于通过 Schur 补集的递归块对角化的分层求解器相结合。这将专门利用由椭圆偏微分方程的强制奇异积分算子产生的分层非对角低秩结构。分层求解器涉及与强制项无关的预计算阶段。一旦这种预计算对算子进行因式分解，许多强迫项的解就具有较低的复杂性，因此只需要一小部分时间。该程序还将考虑在一类重要边界上定义的奇异积分方程：那些是单位间隔和圆的多项式映射。将进行大量分析，以通过谱映射定理再次获得带状奇异积分算子。使用混合边界条件或多重连接轮廓求解奇异积分方程会导致在连接处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充分捕捉连接处出现的复杂奇异结构的底座来解决。新的谱方法将用于解决斯托克斯流、双调和方程以及断裂力学的应力和应变计算问题。新的谱方法与稳定的高阶时间步进方案的结合将允许探索亥姆霍兹方程的时域积分公式和瑞利泰勒不稳定性的模拟。它还将允许在重要应用中进行实验和潜在发现新现象，例如纳米级光学元笼、逆散射问题的解决以及深水中内波的 BenjaminOno 方程的模拟。