Error estimation for quadrature by expansion in layer potential evaluation

Klinteberg, Ludvig af; Tornberg, Anna‐Karin

doi:10.1007/s10444-016-9484-x

Cited by 31 publications

(85 citation statements)

References 12 publications

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“…Specifically, to maintain high-order quadrature accuracy, the expansion center must be placed sufficiently far from the surface element spawning the center as well as nearby elements, where the critical distance depends on the 'size' of the element. As an example providing quantitative detail, the following result, due to af Klinteberg and Tornberg [18], gives the asymptotic error for the case of a smooth tensor product rule over a flat 2h × 2h panel.…”

Section: First Stage: Formation Of a Truncated Local Expansionmentioning

confidence: 99%

Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions

Wala

Klöckner

2020

Journal of Computational Physics

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We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX ('GIGAQBX') and target-specific expansions. GIGAQBX is a fast high-order scheme for evaluation of layer potentials based on Quadrature by Expansion ('QBX') using local expansions formed via the Fast Multipole Method (FMM). Targetspecific expansions serve to lower the cost of local expansion evaluation, reducing the computational effort from O((p + 1) 2 ) to O(p + 1) in three dimensions, without any accuracy loss compared with conventional expansions, but with the loss of source/target separation in the expansion coefficients. GIGAQBX is a 'global' QBX scheme, meaning that the potential is mediated entirely through expansions for points close to or on the boundary. In our scheme, this single global expansion is decomposed into two parts that are evaluated separately: one part incorporating near-field contributions using target-specific expansions, and one part using conventional spherical harmonic expansions of far-field contributions, noting that convergence guarantees only exist for the sum of the two sub-expansions. By contrast, target-specific expansions were originally introduced as an acceleration mechanism for 'local' QBX schemes, in which the far-field does not contribute to the QBX expansion. Compared with the unmodified GIGAQBX algorithm, we show through a reproducible, time-calibrated cost model that the combined scheme yields a considerable cost reduction for the near-field evaluation part of the computation. We support the effectiveness of our scheme through numerical results demonstrating performance improvements for Laplace and Helmholtz kernels. * wala1@illinois.edu † andreask@illinois.edu arXiv:1811.01110v2 [math.NA] 5 Aug 2019 stage-1 node QBX center Figure 1: Depiction of QBX center placement for a discretization over a triangular element. The QBX centers for on-surface evaluation are spawned in the normal direction at the discretization nodes termed 'stage-1' nodes in [32], which serve as targets for on-surface evaluation. See also Section 2.5.While Π is a generally accurate approximation of Sµ for target points away from the source geometry Γ, accuracy substantially decreases in the region close to Γ or on Γ itself. A recent approach to resolving this problem is based on Quadrature by Expansion (QBX, [20]). By using the broadly applicable assumption that the underlying kernel is a (locally) analytic function of the target point x, and leveraging the strengths of smooth high-order quadrature rules for one-and two-dimensional functions, QBX achieves high-order accuracy without sacrificing generality. The key idea behind QBX is that, to extend the applicability of smooth high-order quadrature to all points x ∈ R 3 , the potential (2) can be expanded in a local expansion about a center c ∈ R 3 \ Γ, and this expansion may recover the value of the potential by analytic continuation in regions where smooth quadrature is not adequate. A pote...

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Section: First Stage: Formation Of a Truncated Local Expansionmentioning

confidence: 99%

Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions

Wala

Klöckner

2020

Journal of Computational Physics

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“…This stationary problem is the result of a semi-implicit discretization of the Navier-Stokes equations (3). Because of the divergence-free condition, these equations, like the Navier-Stokes equations, imply a compatibility condition on f , namely ∂Ω f ·n dS y = 0 .…”

Section: Formulationmentioning

confidence: 99%

“…The integral equation approach that we have outlined so far is based on a first order IMEX discretization of the Navier-Stokes equations (3). In order to increase the temporal order of accuracy, we use the semiimplicit spectral deferred corrections (SISDC) method [47].…”

Section: Time Steppingmentioning

confidence: 99%

“…For t * ∈ C such that z = γ(t * ), the quadrature error at z is approximately proportional to ρ(t * ) −2n , where ρ(t) = t ± √ t 2 − 1 , with the sign defined such that ρ > 1, is the Bernstein radius. Even though it is possible to derive more accurate quadrature estimates for the singularities of our kernel (96) (see [3,4]), it is for our purposes sufficient to heuristically determine a limiting radius R, and then apply kernel split quadrature for points z such that ρ(γ −1 (z)) < R. Note that finding t * = γ −1 (z) is a cheap operation, using the Newton-based method of [4].…”

Section: Determining Where To Apply Kernel-split Quadraturementioning

confidence: 99%

“…||u (3) u|| ||u (3) u|| 2 ( t 3 ) ||u (4) u|| ||u (4) u|| 2 ( t 4 ) Figure 8: Time convergence for spin down problem,ũ (K) denotes solution computed using SISDC order K. For small ∆t, the lowest attainable error can be seen to increase with decreasing ∆t. This is related to the results in fig.…”

Section: Time Convergencementioning

confidence: 99%

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A fast integral equation method for the two-dimensional Navier-Stokes equations

Klinteberg¹,

Askham²,

Kropinski³

2020

Journal of Computational Physics

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The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary conditions are handled naturally, and the ill-conditioning caused by high order terms in the PDE is preconditioned analytically. Despite these advantages, the adoption of integral equation methods has been slow due to a number of difficulties in their implementation. This work describes a complete integral equation-based flow solver that builds on recently developed methods for singular quadrature and the solution of PDEs on complex domains, in combination with several more well-established numerical methods. We apply this solver to flow problems on a number of geometries, both simple and challenging, studying its convergence properties and computational performance. This serves as a demonstration that it is now relatively straightforward to develop a robust, efficient, and flexible Navier-Stokes solver, using integral equation methods.

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Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries

Bagge

Tornberg

2021

Numerical Methods in Fluids

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Boundary integral methods are highly suited for problems with complicated geometries, but require special quadrature methods to accurately compute the singular and nearly singular layer potentials that appear in them. This article presents a boundary integral method that can be used to study the motion of rigid particles in three-dimensional periodic Stokes flow with confining walls. A centerpiece of our method is the highly accurate special quadrature method, which is based on a combination of upsampled quadrature and quadrature by expansion, accelerated using a precomputation scheme. The method is demonstrated for rodlike and spheroidal particles, with the confining geometry given by a pipe or a pair of flat walls. A parameter selection strategy for the special quadrature method is presented and tested. Periodic interactions are computed using the spectral Ewald fast summation method, which allows our method to run in O(n log n) time for n grid points in the primary cell, assuming the number of geometrical objects grows while the grid point concentration is kept fixed. K E Y W O R D Sboundary integral equations, fast Ewald summation, quadrature by expansion, rigid particle suspensions, Stokes flow, streamline computation INTRODUCTIONMicrohydrodynamics is the study of fluid flow at low Reynolds numbers, also known as Stokes flow or creeping flow. Applications are found in biology, for example in the swimming of microorganisms 1 and in blood flow, 2 as well as in the field of microfluidics, which concerns the design and construction of miniaturized fluid devices. 3 Suspensions of rigid particles in Stokes flow are important both in various applications and in fundamental fluid mechanics. [4][5][6][7] In this article, we describe a boundary integral method that can be used to study the motion of rigid particles of different shapes in Stokes flow. The particle suspension may also be confined in a container geometry, such as a pipe or a pair of flat walls. The flow in the fluid domain (i.e., within the container but outside the particles) is governed by the Stokes equations, which for an incompressible Newtonian fluid take the formThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Error estimation for quadrature by expansion in layer potential evaluation

Cited by 31 publications

References 12 publications

Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions

Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions

A fast integral equation method for the two-dimensional Navier-Stokes equations

Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries

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