×

A boundary integral equation approach to computing eigenvalues of the Stokes operator. (English) Zbl 1434.76031

Summary: The eigenvalues and eigenfunctions of the Stokes operator have been the subject of intense analytical investigation and have applications in the study and simulation of the Navier-Stokes equations. As the Stokes operator is second order and has the divergence-free constraint, computing these eigenvalues and the corresponding eigenfunctions is a challenging task, particularly in complex geometries and at high frequencies. The boundary integral equation (BIE) framework provides robust and scalable eigenvalue computations due to (a) the reduction in the dimension of the problem to be discretized and (b) the absence of high-frequency “pollution” when using Green’s function to represent propagating waves. In this paper, we detail the theoretical justification for a BIE approach to the Stokes eigenvalue problem on simply- and multiply-connected planar domains, which entails a treatment of the uniqueness theory for oscillatory Stokes equations on exterior domains. Then, using well-established techniques for discretizing BIEs, we present numerical results which confirm the analytical claims of the paper and demonstrate the efficiency of the overall approach.

MSC:

76D07 Stokes and related (Oseen, etc.) flows
45B05 Fredholm integral equations
65R20 Numerical methods for integral equations

Software:

Chebfun; MPSpack; FLAM
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akhmetgaliyev, E.; Bruno, OP; Nigam, N., A boundary integral algorithm for the laplace dirichlet-neumann mixed eigenvalue problem, J. Comput. Phys., 298, 1-28 (2015) · Zbl 1349.65600 · doi:10.1016/j.jcp.2015.05.016
[2] Alpert, BK, Hybrid gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20, 5, 1551-1584 (1999) · Zbl 0933.41019 · doi:10.1137/S1064827597325141
[3] Antunes, PR, On the buckling eigenvalue problem, J. Phys. A: Math. Theor., 44, 21, 215205 (2011) · Zbl 1219.35146 · doi:10.1088/1751-8113/44/21/215205
[4] Ashbaugh, MS; Laugesen, RS, Fundamental tones and buckling loads of clamped plates, Ann. della Scuola Normal. Super. Pisa-Classe Sci., 23, 2, 383-402 (1996) · Zbl 0891.73028
[5] Babuska, IM; Sauter, SA, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal., 34, 6, 2392-2423 (1997) · Zbl 0894.65050 · doi:10.1137/S0036142994269186
[6] Bäcker, A.: Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems. In: The Mathematical Aspects of Quantum Maps, pp. 91-144. Springer (2003) · Zbl 1046.81040
[7] Barnett, A., Asymptotic rate of quantum ergodicity in chaotic euclidean billiards, Commun. Pure Appl. Math. A J. Issued Courant Inst. Math. Sci., 59, 10, 1457-1488 (2006) · Zbl 1133.81022 · doi:10.1002/cpa.20150
[8] Barnett, A., Betcke, T.: MPSpack: A MATLAB toolbox to solve Helmholtz PDE, wave scattering, and eigenvalue problems, pp. 2008-2012
[9] Batcho, PF; Karniadakis, GE, Generalized Stokes eigenfunctions: a new trial basis for the solution of incompressible Navier-Stokes equations, J. Comput. Phys., 115, 1, 121-146 (1994) · Zbl 0820.76066 · doi:10.1006/jcph.1994.1182
[10] Biros, G., Ying, L., Zorin, D.: The embedded boundary integral method for the unsteady incompressible Navier-Stokes equations. Tech. Rep. TR2003-838, Courant Institute, New York University. https://cs.nyu.edu/media/publications/TR2003-838.pdf (2002)
[11] Bjørstad, PE; Tjøstheim, BP, High precision solutions of two fourth order eigenvalue problems, Computing, 63, 2, 97-107 (1999) · Zbl 0940.65119 · doi:10.1007/s006070050053
[12] Bornemann, F., On the numerical evaluation of fredholm determinants, Math. Comput., 79, 270, 871-915 (2010) · Zbl 1208.65182 · doi:10.1090/S0025-5718-09-02280-7
[13] Bramble, J.; Payne, L., Pointwise bounds in the first biharmonic boundary value problem, J. Math. Phys., 42, 1-4, 278-286 (1963) · Zbl 0168.37003 · doi:10.1002/sapm1963421278
[14] Bremer, J.; Gimbutas, Z., A nyström method for weakly singular integral operators on surfaces, J. Comput. Phys., 231, 14, 4885-4903 (2012) · Zbl 1245.65177 · doi:10.1016/j.jcp.2012.04.003
[15] Bremer, J.; Gimbutas, Z.; Rokhlin, V., A nonlinear optimization procedure for generalized Gaussian quadratures, SIAM J. Sci. Comput., 32, 4, 1761-1788 (2010) · Zbl 1215.65045 · doi:10.1137/080737046
[16] Bruno, OP; Kunyansky, LA, A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications, J. Comput. Phys., 169, 1, 80-110 (2001) · Zbl 1052.76052 · doi:10.1006/jcph.2001.6714
[17] Carstensen, C.; Gallistl, D., Guaranteed lower eigenvalue bounds for the biharmonic equation, Numer. Math., 126, 1, 33-51 (2014) · Zbl 1298.65165 · doi:10.1007/s00211-013-0559-z
[18] Chen, W.; Lin, Q., Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method, Appl. Math., 51, 1, 73-88 (2006) · Zbl 1164.65489 · doi:10.1007/s10492-006-0006-x
[19] Cheng, H.; Gimbutas, Z.; Martinsson, PG; Rokhlin, V., On the compression of low rank matrices, SIAM J. Sci. Comput., 26, 4, 1389-1404 (2005) · Zbl 1083.65042 · doi:10.1137/030602678
[20] Colton, DL; Kress, R., Integral equation methods in scattering theory. Pure and Applied Mathematics (1983), New York: Wiley, New York · Zbl 0522.35001
[21] Driscoll, T. A., Hale, N., Trefethen, L.N.: Chebfun guide (2014)
[22] Epstein, CL; O’Neil, M., Smoothed corners and scattered waves, SIAM J. Sci. Comput., 38, 5, A2665-A2698 (2016) · Zbl 1347.65039 · doi:10.1137/15M1028248
[23] Filoche, M.; Mayboroda, S., Strong localization induced by one clamped point in thin plate vibrations, Phys. Rev. Lett., 103, 25, 254301 (2009) · doi:10.1103/PhysRevLett.103.254301
[24] Halko, N.; Martinsson, PG; Tropp, JA, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53, 2, 217-288 (2011) · Zbl 1269.65043 · doi:10.1137/090771806
[25] Helsing, J.; Jiang, S., On integral equation methods for the first Dirichlet problem of the biharmonic and modified biharmonic equations in nonsmooth domains, SIAM J. Sci. Comput., 40, 4, A2609-A2630 (2018) · Zbl 1398.65353 · doi:10.1137/17M1162238
[26] Helsing, J.; Ojala, R., Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning, J. Comput. Phys., 227, 20, 8820-8840 (2008) · Zbl 1152.65114 · doi:10.1016/j.jcp.2008.06.022
[27] Helsing, J.; Ojala, R., On the evaluation of layer potentials close to their sources, J. Comput. Phys., 227, 5, 2899-2921 (2008) · Zbl 1135.65404 · doi:10.1016/j.jcp.2007.11.024
[28] Ho, K.L.: FLAM: Fast linear algebra in MATLAB. 10.5281/zenodo.1253582 (2018)
[29] Ho, KL; Greengard, L., A fast direct solver for structured linear systems by recursive skeletonization, SIAM J. Sci. Comput., 34, 5, A2507-A2532 (2012) · Zbl 1259.65062 · doi:10.1137/120866683
[30] Huang, P., He, Y., Feng, X.: Numerical investigations on several stabilized finite element methods for the Stokes eigenvalue problem. Mathematical Problems in Engineering 2011 (2011) · Zbl 1235.74286
[31] Jia, S.; Xie, H.; Yin, X.; Gao, S., Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods, Appl. Math., 54, 1, 1-15 (2009) · Zbl 1212.65434 · doi:10.1007/s10492-009-0001-0
[32] Jiang, S.; Kropinski, MC; Quaife, BD, Second kind integral equation formulation for the modified biharmonic equation and its applications, J. Comput. Phys., 249, 113-126 (2013) · Zbl 1427.76206 · doi:10.1016/j.jcp.2013.04.034
[33] Johnson, CP; Will, KM, Beam buckling by finite element procedure, J. Struc. Div., 100, Proc Paper 10432, 669-685 (1974)
[34] Kelliher, J., Eigenvalues of the Stokes operator versus the Dirichlet laplacian in the plane, Pac. J. Math., 244, 1, 99-132 (2009) · Zbl 1185.35152 · doi:10.2140/pjm.2010.244.99
[35] Kim, S., Karrila, S.J.: Microhydrodynamics: Principles and selected applications (1991)
[36] Kimeswenger, A.; Steinbach, O.; Unger, G., Coupled finite and boundary element methods for fluid-solid interaction eigenvalue problems, SIAM J. Numer. Anal., 52, 5, 2400-2414 (2014) · Zbl 1307.74032 · doi:10.1137/13093755x
[37] Kirsch, A., An integral equation approach and the interior transmission problem for maxwell’s equations, Inver. Probl. Imaging, 1, 1, 159 (2007) · Zbl 1129.35080 · doi:10.3934/ipi.2007.1.159
[38] Kitahara, M.: Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates, vol. 10 Elsevier (2014) · Zbl 0645.73037
[39] Klöckner, A.; Barnett, A.; Greengard, L.; O’Neil, M., Quadrature by expansion: a new method for the evaluation of layer potentials, J. Comput. Phys., 252, 332-349 (2013) · Zbl 1349.65094 · doi:10.1016/j.jcp.2013.06.027
[40] Kress, R., Boundary integral equations in time-harmonic acoustic scattering, Math. Comput. Model., 15, 3-5, 229-243 (1991) · Zbl 0731.76077 · doi:10.1016/0895-7177(91)90068-I
[41] Kress, R., Maz’ya, V., Kozlov, V.: Linear integral equations, vol. 17. Springer (1989) · Zbl 0671.45001
[42] Ladyzhenskaya, O. A.: The mathematical theory of viscous incompressible flow, vol. 76. Gordon and Breach, New York (1969) · Zbl 0184.52603
[43] Leriche, E.; Labrosse, G., Stokes eigenmodes in square domain and the stream function-vorticity correlation, J. Comput. Phys., 200, 2, 489-511 (2004) · Zbl 1115.76319 · doi:10.1016/j.jcp.2004.03.017
[44] Lindsay, AE; Quaife, B.; Wendelberger, L., A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points, Adv. Comput. Math., 44, 4, 1249-1273 (2018) · Zbl 1516.35285 · doi:10.1007/s10444-017-9580-6
[45] Lovadina, C.; Lyly, M.; Stenberg, R., A posteriori estimates for the Stokes eigenvalue problem, Numer. Methods Partial Differ. Equ., 25, 1, 244-257 (2009) · Zbl 1169.65109 · doi:10.1002/num.20342
[46] Lu, YY; Yau, ST, Eigenvalues of the laplacian through boundary integral equations, SIAM J. Matrix Anal. Appl., 12, 3, 597-609 (1991) · Zbl 0742.65075 · doi:10.1137/0612046
[47] Mayergoyz, ID; Fredkin, DR; Zhang, Z., Electrostatic (plasmon) resonances in nanoparticles, Phys. Rev. B, 72, 15, 155412 (2005) · doi:10.1103/PhysRevB.72.155412
[48] Mercier, B.; Osborn, J.; Rappaz, J.; Raviart, PA, Eigenvalue approximation by mixed and hybrid methods, Math. Comput., 36, 154, 427-453 (1981) · Zbl 0472.65080 · doi:10.1090/S0025-5718-1981-0606505-9
[49] Noferini, V.; Pérez, J., Chebyshev rootfinding via computing eigenvalues of colleague matrices: when is it stable?, Math. Comput., 86, 306, 1741-1767 (2017) · Zbl 1361.65029 · doi:10.1090/mcom/3149
[50] Osborn, JE, Approximation of the eigenvalues of a nonselfadjoint operator arising in the study of the stability of stationary solutions of the Navier-Stokes equations, SIAM J. Numer. Anal., 13, 2, 185-197 (1976) · Zbl 0334.76010 · doi:10.1137/0713019
[51] Overvelde, JT; Shan, S.; Bertoldi, K., Compaction through buckling in 2d periodic, soft and porous structures: effect of pore shape, Adv. Mater., 24, 17, 2337-2342 (2012) · doi:10.1002/adma.201104395
[52] Pólya, G., Pólya, G., Szegő, G.: Isoperimetric inequalities in mathematical physics. Princeton University Press (1951) · Zbl 0044.38301
[53] Pozrikidis, C.: Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press, Cambridge. 10.1017/CBO9780511624124. http://ebooks.cambridge.org/ref/id/CBO9780511624124 (1992) · Zbl 0772.76005
[54] Rachh, M.; Askham, T., Integral equation formulation of the biharmonic Dirichlet problem, J. Sci. Comp., 75, 2, 762-781 (2018) · Zbl 1391.31004 · doi:10.1007/s10915-017-0559-8
[55] Rachh, M., Serkh, K.: On the solution of S,tokes equation on regions with corners. arXiv:1711.04072 (2017) · Zbl 1467.35262
[56] Rannacher, R., Nonconforming finite element methods for eigenvalue problems in linear plate theory, Numer. Math., 33, 1, 23-42 (1979) · Zbl 0394.65035 · doi:10.1007/BF01396493
[57] Reed, M., Simon, B.: Methods of mathematical physics i: Functional analysis (1972) · Zbl 0242.46001
[58] Schneider, K.; Farge, M., Final states of decaying 2d turbulence in bounded domains: Influence of the geometry, Physica D: Nonlinear Phenom., 237, 14-17, 2228-2233 (2008) · Zbl 1143.76469 · doi:10.1016/j.physd.2008.02.012
[59] Serkh, K.; Rokhlin, V., On the solution of elliptic partial differential equations on regions with corners, J. Comput. Phys., 305, 150-171 (2016) · Zbl 1349.35049 · doi:10.1016/j.jcp.2015.10.024
[60] Simon, B.: Trace ideals and their applications, vol. 120 American Mathematical Soc (2005) · Zbl 1074.47001
[61] Steinbach, O.; Unger, G., Convergence analysis of a galerkin boundary element method for the dirichlet laplacian eigenvalue problem, SIAM J. Numer. Anal., 50, 2, 710-728 (2012) · Zbl 1250.65136 · doi:10.1137/100801986
[62] Szegö, G., On membranes and plates, Proc. Natl. Acad. Sci. U.S.A., 36, 3, 210 (1950) · Zbl 0039.20604 · doi:10.1073/pnas.36.3.210
[63] Taylor, G., The buckling load for a rectangular plate with four clamped edges, ZAMM-J. Appl. Math. Mech./Z. Angewandte Math. Mech., 13, 2, 147-152 (1933) · JFM 59.0745.03 · doi:10.1002/zamm.19330130222
[64] Trefethen, L.N.: Approximation theory and approximation practice, vol. 128. SIAM (2013) · Zbl 1264.41001
[65] Trefethen, LN; Betcke, T., Computed eigenmodes of planar regions, Contemp. Math., 412, 297-314 (2006) · Zbl 1107.65101 · doi:10.1090/conm/412/07783
[66] Türeci, HE; Schwefel, HG, An efficient fredholm method for the calculation of highly excited states of billiards, J. Phys. A Math. Theor., 40, 46, 13869 (2007) · Zbl 1125.70016 · doi:10.1088/1751-8113/40/46/004
[67] Veble, G.; Prosen, T.; Robnik, M., Expanded boundary integral method and chaotic time-reversal doublets in quantum billiards, New J. Phys., 9, 1, 15 (2007) · doi:10.1088/1367-2630/9/1/015
[68] Zhao, L.; Barnett, A., Robust and efficient solution of the drum problem via nystrom approximation of the fredholm determinant, SIAM J. Numer. Anal., 53, 4, 1984-2007 (2015) · Zbl 1327.65230 · doi:10.1137/140973992
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.