A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems

Zhang, Yabin; Gillman, Adrianna

doi:10.1007/s10543-020-00818-z

Cited by 6 publications

(31 citation statements)

References 51 publications

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“…Theorem 5.2 states that the proposed spectral Galerkin method has a similar performance to the Nyström one. Indeed, if interfaces belong to C 8 , one obtains super-algebraic convergence for both methods (see [46] for Nyström). The super-algebraic convergence rate of the Nyström method for the transmission problem on a bounded object in two dimensions was rigorously proved in [12].…”

Section: Discrete Problemmentioning

confidence: 99%

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Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

2021

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We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequen- cies (also known as Rayleigh-Wood frequencies), we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nystr\"om methods.

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Section: Discrete Problemmentioning

confidence: 99%

“…Our discretization method employs a quasi-periodic basis so that techniques forcing the quasi-periodicity of the discrete solutions are not necessary (cf. [24,46]). Instead, an accurate approximation of the quasi-periodic Green's function is required in order to extract its Fourier coefficients through the fast Fourier transform (FFT).…”

Section: Introductionmentioning

confidence: 99%

Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

2021

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“…Woodbury formulas such as (12) are well-known in the linear algebra literature [24] and have been the cornerstone of recently developed fast direct solvers for applications including to periodic Stokes flow [1] and quasi-periodic scattering problems [25,26]. While the Woodbury formulas have been used in these applications, it was done so without any concern for the stability of the approach.…”

Section: Stability Of the Woodbury Formulamentioning

confidence: 99%

“…Due to the large number of discretization points on Γ far k , it is often too expensive to build the ID directly. Instead, we organize the discretization points on Γ far k into special structure such as the dyadic partition (See section 3 of [26]) or binary tree (Such as the binary tree used in the HBS forward compression) based upon their distance to Γ p . Then an ID for A far kp is constructed by first building IDs for interaction between points in each individual partition subset or tree node and points on P bas , which corresponds to row subblocks of A far kp .…”

Section: Low-rank Approximation For a Kp A Kc And A Pkmentioning

confidence: 99%

A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

Zhang¹,

Gillman²,

Veerapaneni³

2021

Preprint

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In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, and solving the problem on the refined discretization via a Woodbury formula. Numerical results illustrate the efficiency of the solver in accelerating particulate Stokes simulations.

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“…The multilayered media problem (1) can be transformed into a collection of integral equations defined on each interface via the formulation in [1,6]. This formulation is robust even at Wood's anomalies and is amenable to fast direct solvers.…”

Section: Introductionmentioning

confidence: 99%

A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems, Part II

Zhang¹,

Gillman²

2022

Preprint

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This manuscript is the second in a series presenting fast direct solution techniques for solving twodimensional wave scattering problems from quasi-periodic multilayered structures. The fast direct solvers presented in the series are for the linear system that results from the discretization of a robust integral formulation. The fast direct solver presented in this manuscript has a computational cost that scales linearly with respect to the number of discretization points on the interfaces and the number of layers. The latter is an improvement over the previous solver and makes the new solver more efficient especially for problems involving multiple incident angles and changes to the layered media. Numerical results illustrate the improved performance of the new solver over the previous one.

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A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems

Cited by 6 publications

References 51 publications

Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems, Part II

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