Fikl, Alexandru; Bodony, Daniel J. Jump relations of certain hypersingular Stokes kernels on regular surfaces. (English) Zbl 1452.31008 SIAM J. Appl. Math. 80, No. 5, 2226-2248 (2020). Summary: The jump relations of certain hypersingular Stokes kernels arising from the single-layer potential representation of the velocity field are derived. We find that the jumps in the normal gradients of pressure and stress and the normal component of the velocity Hessian involve the mean curvature and tangential derivatives of the layer potential density. The analysis is performed separately on the normal and tangential components of each kernel and reveals the behavior near the singularity in these scalar kernels as well. Cited in 2 Documents MSC: 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) 45L05 Theoretical approximation of solutions to integral equations Keywords:Stokes equations; hypersingular integral equations; regular surface; jump relations Software:GitHub; pytential PDFBibTeX XMLCite \textit{A. Fikl} and \textit{D. J. Bodony}, SIAM J. Appl. Math. 80, No. 5, 2226--2248 (2020; Zbl 1452.31008) Full Text: DOI References: [1] A. H. Aliabadi, The Boundary Element Method Volume II: Applications in Solids and Structures, John Wiley and Sons, New York, 2002. [2] F. Alouges, A. DeSimone, and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers, Math. Models Methods Appl. Sci., 21 (2011), pp. 361-387. · Zbl 1315.65100 [3] B. K. Alpert, Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20 (1999), pp. 1551-1584. · Zbl 0933.41019 [4] M. P. Brandao, Improper integrals in theoretical aerodynamics: The problem revisited, AIAA J., 25 (1987), pp. 1258-1260, https://doi.org/10.2514/3.9775. [5] M. Carley, Numerical quadratures for singular and hypersingular integrals in boundary element methods, SIAM J. Sci. Comput., 29 (2006), pp. 1207-1216, https://doi.org/10.1137/060666093. · Zbl 1141.65393 [6] A. T. Chwang, Hydromechanics of low-Reynolds-number flow. Motion of a spheroidal particle in quadratic flows, J. Fluid Mech., 72 (1975), pp. 17-34. · Zbl 0331.76062 [7] R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops and Particles, Academic Press, New York, 1978. [8] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed., Springer, New York, 2013. · Zbl 1266.35121 [9] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, SIAM, Philadelphia, 2011. · Zbl 1251.49001 [10] M. P. do Carmo, Differential Geometry of Curves and Surfaces, 2nd ed., Dover Publications, Mienola, NY, 2016. · Zbl 1352.53002 [11] A. Frangi and M. Guiggiani, Free terms and compatibility conditions for \textup3D hypersingular boundary integral equations, ZAMM Z. Angew. Math. Mech., 81 (2001), pp. 651-664, https://doi.org/10.1002/1521-4001(200110)81:10<651::AID-ZAMM651>3.0.CO;2-E. · Zbl 0991.65128 [12] J. B. Freund, Numerical simulation of flowing blood cells, Annu. Rev. Fluid Mech., 46 (2014), pp. 67-95, https://doi.org/10.1146/annurev-fluid-010313-141349. · Zbl 1297.76198 [13] Z. Gao, Y. Ma, and H. Zhuang, Shape optimization for Stokes flow, Appl. Numer. Math., 58 (2008), pp. 827-844. · Zbl 1147.49035 [14] O. Gonzalez, On stable, complete, and singularity-free boundary integral formulations of exterior Stokes flow, SIAM J. Appl. Math., 69 (2009), pp. 933-958, https://doi.org/10.1137/070698154. · Zbl 1173.31002 [15] O. Gonzalez, A theorem on the surface traction field in potential representations of Stokes flow, SIAM J. Appl. Math., 75 (2015), pp. 1578-1598, https://doi.org/10.1137/140978119. · Zbl 1320.31012 [16] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325-348, https://doi.org/10.1016/0021-9991(87)90140-9. · Zbl 0629.65005 [17] L. Greengard and V. Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numer., 6 (1997), pp. 229-269, https://doi.org/10.1017/S0962492900002725. · Zbl 0889.65115 [18] M. Guiggiani, Hypersingular boundary integral equations have an additional free term, Comput. Mech., 16 (1995), pp. 245-248, https://doi.org/10.1007/BF00369869. · Zbl 0840.65117 [19] N. A. Gumerov and R. Duraiswami, Fast multipole methods on graphics processors, J. Comput. Phys., 227 (2008), pp. 8290-8313, https://doi.org/10.1016/j.jcp.2008.05.023. · Zbl 1147.65012 [20] K. L. Ho and L. Greengard, A fast direct solver for structured linear systems by recursive skeletonization, SIAM J. Sci. Comput., 34 (2012), pp. 2507-2532, https://doi.org/10.1137/120866683. · Zbl 1259.65062 [21] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer, Berlin, 2008. · Zbl 1157.65066 [22] M. S. Ingber and L. A. Mondy, Direct second-kind boundary integral formulation for Stokes flow problems, Comput. Mech., 11 (1993), pp. 11-27, https://doi.org/10.1007/BF00370070. · Zbl 0762.76062 [23] S. Jiang, Jump relations of the quadruple layer potential on a regular surface in three dimensions, Appl. Comput. Harmon. Anal., 21 (2006), pp. 395-403, https://doi.org/10.1016/j.acha.2006.03.001. · Zbl 1106.31005 [24] A. Klockner, pytential, 2019, https://github.com/inducer/pytential. [25] A. Klockner, A. Barnett, L. Greengard, and M. O’Neil, Quadrature by expansion: A new method for the evaluation of layer potentials, J. Comput. Phys., 252 (2013), pp. 332-349, https://doi.org/10.1016/j.jcp.2013.06.027. · Zbl 1349.65094 [26] P. Kolm, S. Jiang, and V. Rokhlin, Quadruple and octuple layer potentials in two dimensions I: Analytical apparatus, Appl. Comput. Harmon. Anal., 1 (2003), pp. 47-74, https://doi.org/10.1016/S1063-5203(03)00004-6. · Zbl 1139.35397 [27] R. Kress, Linear Integral Equations, 3rd ed., Springer, New York, 2014. · Zbl 1328.45001 [28] E. Lac and G. M. Homsy, Axisymmetric deformation and stability of a viscous drop in a steady electric field, J. Fluid Mech., 590 (2007), pp. 239-264, https://doi.org/10.1017/S0022112007007999. · Zbl 1141.76481 [29] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd rev. ed., Math. Appl. 2, Gordon and Breach, New York, 1969. · Zbl 0184.52603 [30] S. Li, M. E. Mear, and L. Xiao, Symmetric weak-form integral equation method for three-dimensional fracture analysis, Comput. Methods Appl. Mech. Engrg., 151 (1998), pp. 435-459, https://doi.org/10.1016/S0045-7825(97)00199-0. · Zbl 0906.73074 [31] V. Mantic and F. Paris, Existence and evaluation of the two free terms in the hypersingular boundary integral equation of potential theory, Eng. Anal. Bound. Elem., 16 (1995), pp. 253-260, https://doi.org/10.1016/0955-7997(95)00069-0. [32] P. A. Martin and F. J. Rizzo, Hypersingular integrals: How smooth must the density be?, Int. J. Numer. Meth. Engng., 39 (1996), pp. 687-704, https://doi.org/10.1002/(SICI)1097-0207(19960229)39:4<687::AID-NME876>3.0.CO;2-S. · Zbl 0846.65070 [33] P. G. Martinsson and V. Rokhlin, A fast direct solver for boundary integral equations in two dimensions, J. Comput. Phys., 205 (2005), pp. 1-23, https://doi.org/10.1016/j.jcp.2004.10.033. · Zbl 1078.65112 [34] Y. Mi and M. H. Aliabadi, Dual boundary element method for three-dimensional fracture mechanics analysis, Eng. Anal. Bound. Elem., 10 (1992), pp. 161-171, https://doi.org/10.1016/0955-7997(92)90047-B. [35] B. Mohammadi and O. Pironneau, Applied shape optimization for fluids, Oxford University Press, Oxford, 2010. · Zbl 0970.76003 [36] G. Monegato, Definitions, properties and applications of finite-part integrals, J. Comput. Appl. Math., 229 (2009), pp. 425-439, https://doi.org/10.1016/j.cam.2008.04.006. · Zbl 1166.65061 [37] C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, Cambridge, 1992. · Zbl 0772.76005 [38] C. Pozrikidis, Interfacial dynamics for Stokes flow, J. Comput. Phys., 169 (2001), pp. 250-301, https://doi.org/10.1006/jcph.2000.6582. · Zbl 1046.76012 [39] J. J. R. Silva, H. Power, and L. C. Wrobel, A hypersingular integral equation formulation for Stokes’ flow in ducts, Eng. Anal. Bound. Elem., 12 (1993), pp. 185-1993, https://doi.org/10.1016/0955-7997(93)90014-C. [40] A. T. Tornberg and L. Greengard, A fast multipole method for the three-dimensional Stokes equations, J. Comput. Phys., 227 (2008), pp. 1613-1619, https://doi.org/10.1016/j.jcp.2007.06.029. · Zbl 1290.76116 [41] S. K. Veerapaneni, D. Gueyffier, D. Zorin, and G. Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in \textup2D, J. Comput. Phys., 228 (2009), pp. 2334-2353, https://doi.org/10.1016/j.jcp.2008.11.036. · Zbl 1275.76175 [42] S. W. Walker, The Shape of Things: A Practical Guide to Differential Geometry and the Shape Derivative, SIAM, Philadelphia, 2015. · Zbl 1336.53001 [43] P. Yla-Oijala, M. Taskinen, and S. Jarvenpaa, Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods, Radio Sci., 40 (2005), https://doi.org/10.1029/2004RS003169. · Zbl 1244.78016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.