A new spectral method approach for singular integral equations

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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 Rayleigh—Taylor 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 Benjamin—Ono equation for internal waves in deep water.