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REVIEW 3 major objections 5 minor 63 references

Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper develops an adaptive boundary integral method that solves two-dimensional Stokes flow in complex, corner-laden geometries to user-set accuracy, demonstrated at 1e-9 relative error in a 378-corner vascular network.

desk verdict A genuinely useful and probably sound adaptive Nyström BIE scheme for nonsmooth 2D Stokes flow, with the main caveat that the corner-accuracy claims rest on self-convergence and need an independent check. read the letter →

arxiv 1909.00049 v1 pith:RM5U77PD submitted 2019-08-30 math.NA cs.NA

Bowei Wu , Hai Zhu , Alex Barnett , Shravan Veerapaneni This is my paper · ORCID
classification math.NAcs.NA MSC 65N3876D0765R20
keywords StokesflowboundaryintegralequationsNyströmmethodnearlysingularquadraturecloseevaluationadaptivemeshrefinementcornersingularitiesfastmultipole
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper builds a complete numerical pipeline for low-Reynolds-number flow around and through complicated two-dimensional objects: boundary panels are chosen automatically, every boundary integral is evaluated by high-order quadrature even when the evaluation point sits on or arbitrarily near the boundary, and the resulting linear system is solved iteratively. The central claim is that a user can specify an error tolerance and the scheme will meet it without hand-tuning the mesh, with convergence faster than any power law on smooth boundaries and root-exponential convergence (error of order $e^{{-c sqrt N}}$) when corners are present. If true, this makes boundary integral methods practical for microfluidic channels, porous media, and blood-vessel networks, where geometry is complex and surfaces nearly touch. The demonstration case is a 378-corner vascular network solved to relative L2 error 1e-9 with fewer than 200K boundary points.

What carries the argument

The load-bearing object is the close-evaluation panel quadrature for the four contour integrals I_L, I_C, I_H, and I_S (logarithmic, Cauchy, Hadamard, and supersingular) into which the Stokes single- and double-layer potentials are decomposed. For a target near or on a panel, the derived scalar density is first upsampled to twice the panel degree using fine Legendre nodes, then approximated by a polynomial in the complex variable y via a backward-stable solve of a Vandermonde system; the monomial integrals follow from a recurrence whose base case is the Cauchy integral of a constant, evaluated as a logarithm with a rotated branch cut so the cut is pushed behind the panel. This one construction supplies accurate matrix entries for self-interaction, nearly singular interactions between nearby panels, and off-surface evaluation, and it is what makes the adaptive refinement scheme trustworthy.

What would settle it

Take a Stokes Dirichlet problem with a known exact solution on a domain containing a reentrant corner with an unusual angle, disable the corner-grading rule so panels near the corner are not geometrically refined, and then measure the relative L2 error as the polynomial degree increases: if the error stops decreasing once the panel density develops a non-polynomial singularity, the claim that adaptive grading reduces every panel to a polynomial regime is shown to carry the accuracy statement.

Watch

Extended reading notes

Core claim

The paper's claim is that all Stokes layer potentials needed for a Dirichlet velocity problem—single layer, double layer, and the associated pressure and traction fields—can be evaluated to high accuracy at targets on or arbitrarily close to the boundary by expressing them as combinations of four complex contour integrals (logarithmic, Cauchy, Hadamard, and supersingular) and applying one unified panel quadrature to each. On each panel the derived scalar density is approximated by a polynomial in the complex coordinate y; the monomial integrals are computed through a recurrence seeded by a logarithm whose branch cut is rotated to lie behind the panel, which removes the need to distinguish close evaluation from self evaluation. Around corners the panels are geometrically graded, and panels are also refined when boundaries approach one another, so the discretization adapts automatically to a user-set tolerance. The paper verifies in a 378-corner model vascular network that the relative L2 velocity error reaches 1e-9 with 356,580 degrees of freedom, with GMRES converging steadily and the fast multipole method dominating the CPU time.

Load-bearing premise

The entire method assumes that on every panel the unknown density behaves like a low-degree polynomial in the panel's curved coordinate, and that the numerical routine used to fit that polynomial stays accurate even though the fitting equations are extremely ill-conditioned.

Editorial extensions

If this is right

  • A user can hand a complex 2D geometry, including corners and near-touching boundaries, to the solver with a tolerance and receive a solution that meets it; no hand-tuned panel distribution is needed.
  • For smooth boundaries the error converges faster than any power of the number of unknowns, and with corners it converges like exp(-c sqrt N), so a few hundred thousand unknowns can reach 1e-9 even with hundreds of corners.
  • Because layer potential evaluation is accurate arbitrarily close to the boundary, the same machinery supports computing velocity, pressure, and traction fields, including force calculations on closely interacting bodies.
  • For two smooth surfaces separated by a distance d, the adaptive criterion costs about O(log(1/d)) unknowns where a uniformly resolved grid costs O(1/sqrt d), making dense suspensions of nearly touching particles feasible.
  • The near-linear CPU scaling, dominated by the fast multipole application, means the method can serve as the stationary-geometry component of an evolving particulate flow simulation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same contour-integral reduction may carry over to other elliptic kernels with Cauchy-type and supersingular contour integral analogues, such as Helmholtz, elasticity, or biharmonic problems, since the quadrature machinery is built on the four integrals rather than on Stokes specifics.
  • The corner treatment is geometric rather than analytic; introducing a per-corner singular basis or a compression step for corner blocks would likely lower the degrees of freedom per corner and is a natural next step the paper itself flags.
  • A rigorous a posteriori error estimate for the adaptive refinement criteria would let the scheme quote certified tolerances rather than empirically matched ones; the paper demonstrates tolerance matching numerically but does not prove it.
  • In time-dependent particulate flows, one could keep the adaptively paneled fixed channel geometry unchanged while moving particles are handled by global close-evaluation quadrature; whether compressed operator representations can be updated as panels change is an open question the paper poses.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript presents an adaptive, panel-based Nyström boundary integral scheme for the Stokes Dirichlet problem in two dimensions on complex, multiply connected, nonsmooth domains. The Stokes velocity potentials are rewritten in terms of four scalar complex contour integrals (logarithmic, Cauchy, Hadamard, and supersingular), for which close- and self-evaluation quadratures are constructed by polynomial interpolation of the derived scalar densities and exact monomial integration. Self- and close-touching targets are treated by a unified rule with branch-cut rotation. A three-stage adaptive panel refinement algorithm chooses panel degree p and panel lengths from a user tolerance, with geometric corner grading and closeness refinement. The paper reports superalgebraic convergence on a smooth starfish example, root-exponential convergence for the shuriken and multi-polygon examples, and solves a 378-corner vascular network to a reported relative L2 error of 1e-9 with 356,580 unknowns, with GMRES iteration counts and timing breakdowns. Pressure and traction formulas are derived in an appendix and validated on the smooth example.

Significance. If the claims are sustained, this is a useful practical contribution: it directly handles Stokes layer potentials in physical variables, unifies on-surface and near-surface evaluation, provides a user-tolerance-driven adaptive discretization for nonsmooth multiply connected geometries, and includes pressure and traction variants. The strength of the paper is its combination of detailed quadrature derivation, a concrete adaptive algorithm, and extensive numerical experiments with convergence tables and timings for five distinct geometries. The weak points are that the corner-accuracy claims are measured only against a finer run of the same scheme and that the close-evaluation quadrature rests on an ill-conditioned Vandermonde solve whose corner-adjacent behavior is not independently validated; both need to be addressed before the quantitative accuracy claims can be taken at face value.

major comments (3)
  1. [Section 5, Table 1 and Examples 2–4] All reported errors in the nonsmooth examples are measured against the finest-grid solution of the same scheme, as the authors explicitly state ('the exact solution is not known analytically; therefore, we use the finest grid solution as the reference solution'). This makes the claimed 1e-9 relative L2 error in the 378-corner example an internal self-convergence measure, not a verification of true accuracy: a systematic quadrature error that persists across all refinements, for instance in the close-evaluation blocks of Section 3.3, would be invisible. Please add at least one nonsmooth test with an independent reference, such as a wedge or polygon problem with a known exact Stokes solution containing the corner-singular terms, or a comparison against an independent high-accuracy method (for example, RCIP or an over-resolved finite-element benchmark), and report both the self-convergence error and the error against that independent reference.
  2. [Section 3.3.1, Eq. (29) and Section 3.4] The close-evaluation quadrature rests on solving the Vandermonde system V a = tau_tilde, and the matrix block A is formed via the adjoint solve V^T X = P^T. The paper justifies this with the statement, following [27, App. A], that backward stability of MATLAB mldivide suffices for m<50. Backward stability bounds the residual of the solved linear system, not the forward error in A = X^T L when V is exponentially ill-conditioned and the polynomial coefficients are large. Since all near-corner evidence in Section 5 is self-referential, a systematic error in exactly these blocks would be invisible. Please provide a targeted validation: for a panel touching a reentrant corner and a neighboring target panel (or on-surface target near the corner), compare the Cauchy, logarithmic, Hadamard, and supersingular outputs of A against a high-precision direct quadrature (for example, 64-digit arithmetic), and report sensitivity to m = p, 2p, 3p. If the m<50 heuristic does not hold for corner-adjacent panels, the adaptive corner refinement must be modified.
  3. [Title, Section 5 (Table 1 discussion), and Conclusions] The paper claims a linear-scaling scheme and states in the Conclusions that 'CPU time grows linearly with problem size', but its own asymptotic estimate in Section 5 states that the close-evaluation matrix-vector multiplication time grows as O(N^{3/2}), and Table 1 shows the FMM fraction decreasing from 78.5% to 66.2% as epsilon decreases. Thus the total per-iteration cost is not asymptotically linear, and the claim that the scheme is linear-scaling is not supported beyond the tested tolerance range. Please either implement an O(N) close-correction application (for example, by exploiting low-rank or translation-invariant structure) or revise the title and conclusion claims to 'near-linear for moderate tolerances', with the O(N^{3/2}) close-evaluation bottleneck stated in the abstract and conclusions.
minor comments (5)
  1. [Section 3.3.2, Eq. (40)] In the displayed formula for q_k, the intermediate expression contains log(1+x), while the derivation and the subsequent branch-cut form use -1-x; please correct the notation to avoid ambiguity.
  2. [Section 3.3.1, Remark 3] The claim that flipping the sign of phi is robust for corners of arbitrary angles is not supported by an explicit experiment; a small parameter study sweeping the interior corner angle between 0 and 2pi would strengthen the paper.
  3. [Section 3.4] After presenting the adjoint construction for the Cauchy kernel, please state explicitly that the same construction is used for the logarithmic, Hadamard, and supersingular blocks, and define n (the number of targets) consistently in all cases.
  4. [Figure 8] Please add a legend to Figure 8 and state whether the horizontal axis is the number of nodes N or the number of unknowns 2N, since the text refers to 'required number of unknowns' without specifying the convention.
  5. [Section 5, Example 3] The statement 'approximately 800 degrees of freedom per corner' should be checked against the conventions used in Table 1 and Example 4; the ratio 356,580/378 = 943 uses 2N, whereas the text elsewhere sometimes counts N or 2N without consistent definition.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the BIE formulation, contour-integral reductions, and panel quadratures are derived from standard potential theory and geometric error estimates; self-citations are not load-bearing, though the nonsmooth accuracy claim rests on self-convergence rather than an independent solution.

full rationale

The derivation chain is not circular. The Stokes boundary integral equation (5) is the standard combined-field indirect formulation, with well-posedness cited to [40,42]; the reduction of Stokes layer potentials to Laplace/Cauchy/Hadamard/supersingular contour integrals (Sections 2.3-2.4) is derived in the text; and the close-evaluation quadratures (Section 3.3) solve the Vandermonde system (29) only to interpolate the density samples, then evaluate the monomial integrals exactly by recurrences (34), (40), and (44). No parameter is fitted to the target solutions: the adaptive constants (C=2.5, lambda=3, alpha, beta, c'=0.7) are stated empirical heuristics that tune discretization efficiency, not calibrated to reproduce the reported errors. The self-citations to [25] (Barnett, Wu, Veerapaneni) are not load-bearing: equations (19)-(20) are re-derived in the paper, and [25] is used only as a comparison method in Example 5. The cited [27] is external (Helsing-Ojala) and supplies the polynomial-interpolation/recursion idea plus the m<50 backward-stability heuristic; that heuristic is a correctness risk, not a circular input. One limitation should be flagged: Section 5 states, 'In all the examples, the exact solution is not known analytically; therefore, we use the finest grid solution as the reference solution.' Thus the nonsmooth 1e-9 claim is a self-convergence check against the same scheme at epsilon=10^-10, not an independent error bound. This weakens external validation but is not circular derivation: the tolerance epsilon is an input to the adaptive panelizer, and the measured difference to a finer-grid solution is an output, not a quantity defined to equal epsilon. The ill-conditioned Vandermonde blocks in Section 3.3.1 are likewise an assumption about linear-solver backward stability, not a prediction forced by the method's inputs.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The scheme is built on standard BIE theory and on polynomial approximation of densities in the complex plane; the adaptive stage introduces a set of manually tuned constants (C, c, alpha, beta, lambda, c', phi) that are not derived from error analysis. No new physical entities are introduced.

free parameters (7)
  • C (close/far ellipse constant) = 2.5
    Sets the elliptical neighborhood in Eq. (27) that decides when the plain Nyström rule is replaced by close-evaluation; chosen empirically to balance accuracy and cost.
  • c (p estimate offset) = 1
    Extra term in Eq. (45) to account for unknown prefactors in the Bernstein ellipse convergence estimate.
  • alpha (corner grading exponent) = 0.5 to 1.1 depending on corner angle
    Controls how short panels touching a corner become (Algorithm 1, line 6); values are chosen per corner angle and tuned to reduce unknowns while retaining accuracy.
  • lambda (corner grading ratio) = 2 or 3
    Ratio between neighboring panel lengths near a corner; lambda=3 for flat corners, lambda=2 for sharp corners.
  • beta (refinement tolerance exponent) = 0.8 (Example 1) or 1 (default)
    Exponent in the test-function refinement criterion (Algorithm 1, line 17); lower beta reduces N in the smooth example.
  • c' (close-touching refinement constant) = 0.7
    Constant in the modified refinement criterion in Remark 5 for close smooth surfaces; tuned to achieve the requested tolerance.
  • phi (branch cut rotation angle) = ±pi/4
    Rotation used in Eq. (33) to move the branch cut behind the panel; sign chosen by normal direction, magnitude chosen so the cut does not interfere with target points.
assumptions (6)
  • domain assumption Combined-field BIE (5) with S+D admixture is Fredholm of second kind and well-posed for the Stokes Dirichlet problem on multiply connected domains.
    Invoked in Section 2.1 and relies on standard results [40,42]; not proved in the paper.
  • standard math p-point Gauss-Legendre quadrature on a panel converges as O(rho^(-2p)) when the integrand is analytic in the Bernstein ellipse of parameter rho.
    Used in Section 3.2 and Stage 1 of Section 4 to set p in Eq. (45); cited to Trefethen Thm 19.3.
  • domain assumption For the Stokes Dirichlet problem on a corner, the density is a constant plus a bounded singular function with power > 1/2, so geometrically graded panels with alpha near 1 control the singularity.
    Used in Stage 2 (corner refinement) to justify the stopping criterion; cited to [39] and [50].
  • ad hoc to paper The Vandermonde system (29) is solved backward stably by MATLAB mldivide for m<50 despite exponential condition number, so the polynomial interpolant matches nodal values near machine precision.
    The entire close-evaluation scheme in Section 3.3.1 depends on this empirical stability claim, cited to [27, App. A].
  • domain assumption In the narrow gap between two smooth curves separated by distance d, the width of the force or density bump scales as O(sqrt(d/kappa)).
    Used in Remark 5 to justify the looser refinement criterion for close-touching surfaces; cited to [56].
  • domain assumption Rotating the branch cut in Eq. (33) by phi=±pi/4 yields correct values of p1 for all target points in the closure of Omega near the panel, with no winding-number tests.
    Used to unify close and self evaluation; the paper notes it can fail for very curved panels or targets near the back side of the panel, and Remark 3 describes a corner exception.

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Cite this review

Pith. "Pith review of Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme." pith.science (2026). https://pith.science/paper/RM5U77PD

@misc{pith2026190900049,
  author       = {Pith},
  title        = {Pith review of: Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RM5U77PD}},
  note         = {Machine review of arXiv:1909.00049}
}
read the original abstract

We present a fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex---possibly nonsmooth---geometries in two dimensions. The key ingredient is a set of panel quadrature rules capable of evaluating weakly-singular, nearly-singular and hyper-singular integrals to high accuracy. Near-singular integral evaluation, in particular, is done using an extension of the scheme developed in J.~Helsing and R.~Ojala, {\it J. Comput. Phys.} {\bf 227} (2008) 2899--2921. The boundary of the given geometry is ``panelized'' automatically to achieve user-prescribed precision. We show that this adaptive panel refinement procedure works well in practice even in the case of complex geometries with large number of corners. In one example, for instance, a model 2D vascular network with 378 corners required less than 200K discretization points to obtain a 9-digit solution accuracy.

Figures

Figures reproduced from arXiv: 1909.00049 by the authors.

Figure 1
Figure 1. Snapshot from a simulation of bacterial supension flow in a microfluidic chip geometry, which is inspired from the design proposed in [23]. A squirmer model [24] is used for modeling the bacteria, which treats them as rigid bodies with a prescribed slip at the fluid-structure interface. Thereby, we solve the Stokes equations with a no-slip boundary condition on the microfluidic chip geometry, a prescribed tangential… view at source ↗
Figure 2
Figure 2. Solution of the Stokes equation in a nonsmooth circular vascular network with Dirichlet boundary condition. We apply no-slip boundary condition at all branch walls, and it is driven by a uniform flow from inner to outer circle. Color here indicates log of the magnitude of fluid velocity. We used automatically gen￾erated panels for both smooth boundaries and 378 corners, resulting in 356, 580 degrees of freedom. GMRE… view at source ↗
Figure 3
Figure 3. Special handling of close evaluation branch cut when the panel is touching a reentrant corner. (a) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (a) Linear shear flow past a starfish-shaped island. Streamlines of the flow and panel endpoints [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: (a) Linear shear flow around a shuriken-shaped island with 8 corners. Streamlines of the flow and [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: (a) Streamlines of a shear flow past 50 randomly generated polygonal islands with a total number [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Error and timing of solving the vascular network BVP. (a) Convergence of errors in log-linear [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Convergence of a uniform flow past two touching disks that are [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

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Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages

  1. [27]

    Helsing and R

    J. Helsing and R. Ojala. On the evaluation of layer potentials close to their sources. Journal of Computational Physics, 227:2899–2921, 2008

    work page 2008

  2. [1]

    Youngren G. K. and A. Acrivos. Stokes flow past a particle of arbitrary shape: a numerical method of solution. Journal of Fluid Mechanics , 69:377–403, May 1975

    work page 1975

  3. [2]

    Youngren G. K. and A. Acrivos. On the shape of a gas bubble in a viscous extensional flow. Journal of Fluid Mechanics , 76:433–442, August 1976

    work page 1976

  4. [3]

    A highly accurate boundary integral equation method for surfactant-laden drops in 3D

    Chiara Sorgentone and Anna-Karin Tornberg. A highly accurate boundary integral equation method for surfactant-laden drops in 3D. Journal of Computational Physics , 360:167–191, 2018

    work page 2018

  5. [4]

    Boundary integral method for the flow of vesicles with viscosity contrast in three dimensions

    Abtin Rahimian, Shravan K Veerapaneni, Denis Zorin, and George Biros. Boundary integral method for the flow of vesicles with viscosity contrast in three dimensions. Journal of Computational Physics , 298:766–786, 2015

    work page 2015

  6. [5]

    Global stability of flowing red blood cell trains

    Spencer H Bryngelson and Jonathan B Freund. Global stability of flowing red blood cell trains. Physical Review Fluids, 3(7):073101, 2018

    work page 2018

  7. [6]

    Integral equation methods for Stokes flow and isotropic elasticity in the plane

    Leslie Greengard, Mary Catherine Kropinski, and Anita Mayo. Integral equation methods for Stokes flow and isotropic elasticity in the plane. Journal of Computational Physics , 125(2):403–414, 1996

    work page 1996

  8. [7]

    L. Ying, G. Biros, and D. Zorin. A kernel-independent adaptive fast multipole method in two and three dimensions. Journal of Computational Physics , 196(2):591–626, 2004

    work page 2004

Show all 63 references
  1. [8]

    A parallel fast multipole accelerated integral equation scheme for 3D Stokes equations

    Haitao Wang, Ting Lei, Jin Li, Jingfang Huang, and Zhenhan Yao. A parallel fast multipole accelerated integral equation scheme for 3D Stokes equations. International journal for numerical methods in engineering, 70(7):812–839, 2007

    2007

  2. [9]

    A fast multipole method for the three-dimensional Stokes equations

    Anna-Karin Tornberg and Leslie Greengard. A fast multipole method for the three-dimensional Stokes equations. Journal of Computational Physics , 227(3):1613–1619, 2008. 21

    2008

  3. [10]

    Computational software: Simple FMM libraries for electro- statics, slow viscous flow, and frequency-domain wave propagation

    Zydrunas Gimbutas and Leslie Greengard. Computational software: Simple FMM libraries for electro- statics, slow viscous flow, and frequency-domain wave propagation. Communications in Computational Physics, 18(2):516–528, 2015

    2015

  4. [11]

    PVFMM: A parallel kernel independent FMM for particle and volume potentials

    Dhairya Malhotra and George Biros. PVFMM: A parallel kernel independent FMM for particle and volume potentials. Communications in Computational Physics , 18(3):808–830, 2015

    2015

  5. [12]

    A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: The sedimentation of fibers

    David Saintillan, Eric Darve, and Eric SG Shaqfeh. A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: The sedimentation of fibers. Physics of Fluids , 17(3):033301, 2005

    2005

  6. [13]

    Spectrally accurate fast summation for periodic Stokes poten- tials

    Dag Lindbo and Anna-Karin Tornberg. Spectrally accurate fast summation for periodic Stokes poten- tials. Journal of Computational Physics , 229(23):8994–9010, 2010

    2010

  7. [14]

    Accelerated boundary integral method for multiphase flow in non-periodic geometries

    Amit Kumar and Michael D Graham. Accelerated boundary integral method for multiphase flow in non-periodic geometries. Journal of Computational Physics , 231(20):6682–6713, 2012

    2012

  8. [15]

    Fast Ewald summation for free-space Stokes potentials

    Ludvig af Klinteberg, Davoud Saffar Shamshirgar, and Anna-Karin Tornberg. Fast Ewald summation for free-space Stokes potentials. Research in the Mathematical Sciences , 4(1):1, 2017

    2017

  9. [16]

    An O(N ) algorithm for Stokes and Laplace interactions of particles

    Ashok S Sangani and Guobiao Mo. An O(N ) algorithm for Stokes and Laplace interactions of particles. Physics of Fluids , 8(8):1990–2010, 1996

    1990

  10. [17]

    Zinchenko and R.H

    A.Z. Zinchenko and R.H. Davis. An efficient algorithm for hydrodynamical interaction of many de- formable drops. Journal of Computational Physics , 157(2):539–587, 2000

    2000

  11. [18]

    Algorithms in FastStokes and its application to micromachined device simulation

    Xin Wang, Joe Kanapka, Wenjing Ye, Narayan R Aluru, and Jacob White. Algorithms in FastStokes and its application to micromachined device simulation. IEEE Transactions on computer-aided design of integrated circuits and systems , 25(2):248–257, 2006

    2006

  12. [19]

    A treecode algorithm for 3D stokeslets and stresslets

    Lei Wang, Svetlana Tlupova, and Robert Krasny. A treecode algorithm for 3D stokeslets and stresslets. arXiv preprint arXiv:1811.12498 , 2018

    2018 arXiv

  13. [20]

    Petascale direct numerical simulation of blood flow on 200k cores and heterogeneous architectures

    Abtin Rahimian, Ilya Lashuk, Shravan Veerapaneni, Aparna Chandramowlishwaran, Dhairya Malhotra, Logan Moon, Rahul Sampath, Aashay Shringarpure, Jeffrey Vetter, Richard Vuduc, Denis Zorin, and George Biros. Petascale direct numerical simulation of blood flow on 200k cores and het...

    2010

  14. [21]

    A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics

    Ehssan Nazockdast, Abtin Rahimian, Denis Zorin, and Michael Shelley. A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics. Journal of Computational Physics , 329:173– 209, 2017

    2017

  15. [22]

    A scalable computational platform for particulate Stokes suspensions

    Wen Yan, Eduardo Corona, Dhairya Malhotra, Shravan Veerapaneni, and Michael Shelley. A scalable computational platform for particulate Stokes suspensions. under review, 2019

    2019

  16. [23]

    A microfabricated array of clamps for immobilizing and imaging c

    S Elizabeth Hulme, Sergey S Shevkoplyas, Javier Apfeld, Walter Fontana, and George M Whitesides. A microfabricated array of clamps for immobilizing and imaging c. elegans. Lab on a Chip , 7(11):1515– 1523, 2007

    2007

  17. [24]

    The rheology of a semi-dilute suspension of swimming model micro- organisms

    Takuji Ishikawa and TJ Pedley. The rheology of a semi-dilute suspension of swimming model micro- organisms. Journal of Fluid Mechanics , 588:399–435, 2007

    2007

  18. [25]

    Spectrally accurate quadratures for evaluation of layer potentials close to the boundary for the 2D Stokes and Laplace equations

    Alex Barnett, Bowei Wu, and Shravan Veerapaneni. Spectrally accurate quadratures for evaluation of layer potentials close to the boundary for the 2D Stokes and Laplace equations. SIAM Journal on Scientific Computing, 37(4):B519–B542, 2015

    2015

  19. [26]

    An accurate integral equation method for simulating multi- phase Stokes flow

    Rikard Ojala and Anna-Karin Tornberg. An accurate integral equation method for simulating multi- phase Stokes flow. Journal of Computational Physics , 298:145–160, 2015. 22

    2015

  20. [28]

    M. C. A. Kropinski. An efficient numerical method for studying interfacial motion in two-dimensional creeping flows. Journal of Computational Physics , 171(2):479–508, 2001

    2001

  21. [29]

    Quadrature by expansion: A new method for the evaluation of layer potentials

    Andreas Kl¨ ockner, Alexander Barnett, Leslie Greengard, and Michael O’Neil. Quadrature by expansion: A new method for the evaluation of layer potentials. Journal of Computational Physics , 252:332–349, 2013

    2013

  22. [30]

    Evaluation of layer potentials close to the boundary for Laplace and Helmholtz problems on analytic planar domains

    Alex H Barnett. Evaluation of layer potentials close to the boundary for Laplace and Helmholtz problems on analytic planar domains. SIAM Journal on Scientific Computing , 36(2):A427–A451, 2014

    2014

  23. [31]

    Asymptotic analysis for close evaluation of layer potentials

    Camille Carvalho, Shilpa Khatri, and Arnold D Kim. Asymptotic analysis for close evaluation of layer potentials. Journal of Computational Physics , 355:327–341, 2018

    2018

  24. [32]

    Ubiquitous evaluation of layer potentials using quadra- ture by kernel-independent expansion

    Abtin Rahimian, Alex Barnett, and Denis Zorin. Ubiquitous evaluation of layer potentials using quadra- ture by kernel-independent expansion. BIT Numerical Mathematics , 58(2):423–456, 2018

    2018

  25. [33]

    Adaptive quadrature by expansion for layer potential evaluation in two dimensions

    Ludvig af Klinteberg and Anna-Karin Tornberg. Adaptive quadrature by expansion for layer potential evaluation in two dimensions. arXiv preprint arXiv:1704.02219 , 2017

    2017 arXiv

  26. [34]

    Harmonic density interpolation methods for high-order evaluation of laplace layer potentials in 2D and 3D

    Carlos P´ erez-Arancibia, Luiz M Faria, and Catalin Turc. Harmonic density interpolation methods for high-order evaluation of laplace layer potentials in 2D and 3D. Journal of Computational Physics , 376:411–434, 2019

    2019

  27. [35]

    Veerapaneni, Abtin Rahimian, George Biros, and Denis Zorin

    Shravan K. Veerapaneni, Abtin Rahimian, George Biros, and Denis Zorin. A fast algorithm for simu- lating vesicle flows in three dimensions. J. Comput. Phys. , 230(14):5610–5634, 2011

    2011

  28. [36]

    Error estimation and adaptive mesh refinement in boundary element method, an overview

    Eisuke Kita and Norio Kamiya. Error estimation and adaptive mesh refinement in boundary element method, an overview. Engineering Analysis with Boundary Elements , 25(7):479–495, 2001

    2001

  29. [37]

    The hp-version of the boundary element method on polygons

    N Heuer and E P Stephan. The hp-version of the boundary element method on polygons. J. Integral Equ. Appl., 8(2):173–212, 1996

    1996

  30. [38]

    Efficient and accurate implementation of hp-BEM for the Laplace operator in 2D

    Markus Bantle and Stefan Funken. Efficient and accurate implementation of hp-BEM for the Laplace operator in 2D. Applied Numerical Mathematics , 95:51 – 61, 2015. Fourth Chilean Workshop on Numerical Analysis of Partial Differential Equations (WONAPDE 2013)

    2015

  31. [39]

    On the solution of Stokes equation on regions with corners

    Manas Rachh and Kirill Serkh. On the solution of Stokes equation on regions with corners. arXiv preprint arXiv:1711.04072; submitted to Comm. Pure Appl. Math., 2017

    2017 arXiv

  32. [40]

    Boundary integral and singularity methods for linearized viscous flow

    Constantine Pozrikidis. Boundary integral and singularity methods for linearized viscous flow . Cam- bridge University Press, 1992

    1992

  33. [41]

    G. C. Hsiao and W. L. Wendland. Boundary integral equations , volume 164 of Applied Mathematical Sciences. Springer, 2008

    2008

  34. [42]

    F.-K. Hebeker. Efficient boundary element methods for three-dimensional exterior viscous flows. Numer. Methods Partial Differential Equations , 2:273–297, 1986

    1986

  35. [43]

    Linear Integral Equations , volume 82 of Applied Mathematical Sciences

    Rainer Kress. Linear Integral Equations , volume 82 of Applied Mathematical Sciences . Springer, 2nd edition, 1999

    1999

  36. [44]

    bhfmm2d: parallel Fortran code for the biharmonic FMM in 2D, 2012

    M Rachh. bhfmm2d: parallel Fortran code for the biharmonic FMM in 2D, 2012

    2012

  37. [45]

    On the numerical solution of the biharmonic equation in the plane

    A Greenbaum, L Greengard, and A Mayo. On the numerical solution of the biharmonic equation in the plane. Physica D, 60(1–4):216–225, 1992. 23

    1992

  38. [46]

    S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young. High-order accurate Nystr¨ om discretization of integral equations with weakly singular kernels on smooth curves in the plane. Adv. Comput. Math. , 40(1):245–272, 2014

    2014

  39. [47]

    Approximation theory and approximation practice , volume 128

    Lloyd N Trefethen. Approximation theory and approximation practice , volume 128. SIAM, 2013

    2013

  40. [48]

    Adaptive quadrature by expansion for layer potential evaluation in two dimensions

    L af Klinteberg and A-K Tornberg. Adaptive quadrature by expansion for layer potential evaluation in two dimensions. SIAM J. Sci. Comput. , 40(3):A1225—A1249, 2018

    2018

  41. [49]

    How bad are Vandermonde matrices? SIAM J

    V Y Pan. How bad are Vandermonde matrices? SIAM J. Matrix Anal. Appl. , 37(2):676–694, 2016

    2016

  42. [50]

    The asymptotic form of the solutions of the integral equations of po- tential theory in the neighbourhood of the corner points of a contour

    S S Zargaryan and V G Maz’ya. The asymptotic form of the solutions of the integral equations of po- tential theory in the neighbourhood of the corner points of a contour. Prikl. Matem. Mekhan. U.S.S.R. , 48(1):120–124, 1984

    1984

  43. [51]

    kdtree code

    Andrea Tagliasacchi. kdtree code. https://www.mathworks.com/matlabcentral/fileexchange/21512- ataiya-kdtree, 2017

    2017

  44. [52]

    Trefethen

    Abinand Gopal and Lloyd N. Trefethen. Solving Laplace problems with corner singularities via rational functions. SIAM Journal on Numerical Analysis (to appear) , 2019

    2019

  45. [53]

    Genetic determinants of hyaloid and retinal vasculature in zebrafish.BMC developmental biology, 7(1):114, 2007

    Yolanda Alvarez, Maria L Cederlund, David C Cottell, Brent R Bill, Stephen C Ekker, Jesus Torres- Vazquez, Brant M Weinstein, David R Hyde, Thomas S Vihtelic, and Breandan N Kennedy. Genetic determinants of hyaloid and retinal vasculature in zebrafish.BMC developmental biology,...

    2007

  46. [54]

    memorygraph: a MATLAB/octave unix tool to record true memory and CPU usage vs time

    Alex Barnett. memorygraph: a MATLAB/octave unix tool to record true memory and CPU usage vs time. https://github.com/ahbarnett/memorygraph, 2018

    2018

  47. [55]

    A unified integral equation scheme for doubly periodic Laplace and Stokes boundary value problems in two dimensions

    Alex H Barnett, Gary R Marple, Shravan Veerapaneni, and Lin Zhao. A unified integral equation scheme for doubly periodic Laplace and Stokes boundary value problems in two dimensions. Communications on Pure and Applied Mathematics , 71(11):2334–2380, 2018

    2018

  48. [56]

    Inclusion of lubrication forces in dynamic simulations

    Ashok S Sangani and Guobiao Mo. Inclusion of lubrication forces in dynamic simulations. Physics of fluids, 6(5):1653–1662, 1994

    1994

  49. [57]

    Solving integral equations on piecewise smooth boundaries using the rcip method: a tutorial

    Johan Helsing. Solving integral equations on piecewise smooth boundaries using the rcip method: a tutorial. In Abstract and Applied Analysis , volume 2013. Hindawi, 2013

    2013

  50. [58]

    On integral equation methods for the first Dirichlet problem of the biharmonic and modified biharmonic equations in nonsmooth domains

    Johan Helsing and Shidong Jiang. On integral equation methods for the first Dirichlet problem of the biharmonic and modified biharmonic equations in nonsmooth domains. SIAM Journal on Scientific Computing, 40(4):A2609–A2630, 2018

    2018

  51. [59]

    Optimal design of deterministic lateral displacement device for viscosity-contrast-based cell sorting

    G¨ okberk Kabacao˘ glu and George Biros. Optimal design of deterministic lateral displacement device for viscosity-contrast-based cell sorting. Physical Review Fluids, 3(12):124201, 2018

    2018

  52. [60]

    Shape optimization of Stokesian peristaltic pumps using boundary integral methods

    Marc Bonnet, Ruowen Liu, and Shravan Veerapaneni. Shape optimization of Stokesian peristaltic pumps using boundary integral methods. arXiv preprint arXiv:1903.03634 , 2019

    1903 arXiv

  53. [61]

    Barnett, Adrianna Gillman, and Shravan K

    Gary Marple, Alexander H. Barnett, Adrianna Gillman, and Shravan K. Veerapaneni. A fast algorithm for simulating multiphase flows through periodic geometries of arbitrary shape. SIAM Journal on Scientific Computing, 38(5):B740–B772, 2016

    2016

  54. [62]

    A fast direct solver for boundary value problems on locally perturbed geometries

    Yabin Zhang and Adrianna Gillman. A fast direct solver for boundary value problems on locally perturbed geometries. Journal of Computational Physics , 356:356–371, 2018

    2018

  55. [63]

    Fast Multipole Boundary Element Method: Theory and Applications in Engineering

    Y Liu. Fast Multipole Boundary Element Method: Theory and Applications in Engineering . Cambridge University Press, 2009. 24

    2009

Pith tools

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