High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane

Hao, Sijia; Barnett, Alex H.; Martinsson, Per-Gunnar; Young, Patrick

doi:10.1007/s10444-013-9306-3

Cited by 105 publications

(166 citation statements)

References 32 publications

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“…However, there exist several high-order discretizations for such a kernel [2,33,45]; when the N per inclusion is not large we prefer the spectral product quadrature due to Martensen, Kussmaul, and Kress [45, chap. However, there exist several high-order discretizations for such a kernel [2,33,45]; when the N per inclusion is not large we prefer the spectral product quadrature due to Martensen, Kussmaul, and Kress [45, chap.…”

Section: Integral Representation and The Stokes Elsmentioning

confidence: 99%

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

Barnett

Marple

Veerapaneni

et al. 2018

Comm Pure Appl Math

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We present a spectrally accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. The basic idea is to use a small least squares solve to enforce periodic boundary conditions without ever handling periodic Green's functions. We describe fast solvers for the two-dimensional (2D) doubly periodic conduction problem and Stokes nonslip fluid flow problem, where the unit cell contains many inclusions with smooth boundaries. Applications include computing the effective bulk properties of composite media (homogenization) and microfluidic chip design.We split the infinite sum over the lattice of images into a directly summed "near" part plus a small number of auxiliary sources that represent the (smooth) remaining "far" contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank-1 or rank-3 correction and a Schur complement, leaves a well-conditioned square system that can be solved iteratively using fast multipole acceleration plus a low-rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, or particle meshes are required), allows adaptivity, and is essentially dimension-and PDE-independent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle close-to-touching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against highaccuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 10 4 randomly located inclusions per unit cell.

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Section: Integral Representation and The Stokes Elsmentioning

confidence: 99%

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

Barnett

Marple

Veerapaneni

et al. 2018

Comm Pure Appl Math

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“…10) induced by a charge distribution on γ with density ρ : ½−1,1 → C. In other words, let ϕ be defined by [1]. Suppose further that g : ½−1,1 → C is defined by [2].…”

Section: Appendixmentioning

confidence: 99%

“…This environment is well understood; the existence and uniqueness of the solutions follow from Fredholm's theory, and the integral equations can be solved numerically using standard tools (see, for example, ref. 1).…”

mentioning

confidence: 99%

On the solution of the Helmholtz equation on regions with corners

Serkh

Rokhlin

2016

Proc. Natl. Acad. Sci. U.S.A.

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In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.boundary value problems | potential theory | corners | elliptic partial differential equations | Helmholtz equation I n potential theory, the Helmholtz equation is reduced to an integral equation by representing the solutions as single-layer or double-layer Helmholtz potentials on the boundaries of the regions. By taking the limit of the solutions to the boundary, the densities of these potentials are shown to satisfy Fredholm integral equations of the second kind.When the boundaries of the regions are smooth, the kernels of the integral equations are weakly singular, and the solutions are also smooth. This environment is well understood; the existence and uniqueness of the solutions follow from Fredholm's theory, and the integral equations can be solved numerically using standard tools (see, for example, ref. 1).When the boundaries of the regions have perfectly sharp corners, both the kernels and the solutions of the integral equations are singular. The behavior in the vicinity of corners of the solutions of both the integral equations and the underlying differential equation have been the subject of much study (see refs. 2 and 3 for representative examples), although the differential equation appears to have received more attention than the integral equations. Comprehensive reviews of the literature can be found in (for example) refs. 4 and 5.The leading singular terms in the solutions, in the vicinity of corners, to both the integral and differential equations are known (for example, ref. 2), and there are a number of theorems describing the spaces to which the solutions belong (for example, refs. 6 and 7). In 1979, R. J. Riddell published a heuristic argument for the existence of a certain asymptotic series for the solutions to the integral equations near the corners, but this line of investigation does not appear to have been pursued further (8).In this paper, we provide a detailed description of the behavior of the solutions to the integral equations in the vicinity of corners, in the specific case of polygonal boundaries. We observe that the solutions in the vicinity of corners are representable by certain series of appropriately selected Bessel functions. The analytical results are used to construct highly accurate and efficient numerical algorithms and are demonstrated by a number a numerical examples. This paper is based on several specific analytical observations, which are described in the following section.

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“…Our goal is to allow for observations of general qualitative curve dynamic behaviour which do not require sophisticated computational acceleration. However, we have appropriately separated regular and integrable singular terms so that the method could be altered for different quadrature rules such as those in [16]. Higher order spatial discretizations can be implemented at the cost of linear system sparsity.…”

Section: Discussionmentioning

confidence: 99%

“…A recent survey article (cf. [16]) demonstrates other quadrature techniques, some of which do not need interpolation and could be applied to this problem. Of particular interest could be the scheme by Kapur and Rokhlin (cf.…”

Section: Discretizing Integralsmentioning

confidence: 99%

A numerical framework for singular limits of a class of reaction diffusion problems

Moyles

Wetton

2015

Journal of Computational Physics

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We present a numerical framework for solving localized pattern structures of reaction-diffusion type far from the Turing regime. We exploit asymptotic structure in a set of well established pattern formation problems to analyze a singular limit model that avoids time and space adaptation typically associated to full numerical simulations of the same problems. The singular model involves the motion of a curve on which one of the chemical species is concentrated. The curve motion is non-local with an integral equation that has a logarithmic singularity. We generalize our scheme for various reaction terms and show its robustness to other models with logarithmic singularity structures. One such model is the 2D Mullins-Sekerka flow which we implement as a test case of the method. We then analyze a specific model problem, the saturated Gierer-Meinhardt problem, where we demonstrate dynamic patterns for a variety of parameters and curve geometries.

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High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane

Cited by 105 publications

References 32 publications

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

On the solution of the Helmholtz equation on regions with corners

A numerical framework for singular limits of a class of reaction diffusion problems

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