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A fast multi-level boundary element method for the Helmholtz equation. (English) Zbl 1075.76587
Summary: Most recently, we have developed a novel multi-level boundary element method (MLBEM) for the solution of the steady heat diffusion equation involving asymptotically decaying non-oscillatory log-singular and strongly singular kernels. This hierarchical approach generalizes the pioneering work of Brandt and Lubrecht on multi-level multi-integration (MLMI) and C-cycle multi-grid to the broader class of mixed boundary value problems. The result is a fast, accurate and efficient boundary element method. The present paper extends this new computational methodology to the solution of the Helmholtz equation involving oscillatory log-singular and strongly singular kernels for two-dimensional problems. We consider a direct boundary element formulation and, due to the nature of the fundamental solutions, split the corresponding boundary integral equation into real and imaginary parts. Then, we introduce double-noded corners to facilitate a patch-by-patch application of the MLMI algorithm for fast matrix--vector and matrix-transpose--vector multiplications within bi-conjugate gradient methods. The performance of the proposed fast MLBEM is investigated using a numerical example that possesses an exact solution. For wave numbers $\kappa = 20$ and below, we demonstrate that the fast MLBEM algorithm for the Helmholtz equation is robust, accurate, and exceptionally efficient.

76M15Boundary element methods (fluid mechanics)
76Q05Hydro- and aero-acoustics
65N38Boundary element methods (BVP of PDE)
Full Text: DOI
[1] Schenck, H. A.: Improved integral formulation for acoustic radiation problems. J. acoust. Soc. am. 44, 41-58 (1968) · Zbl 0187.50302
[2] Meyer, W. L.; Bell, W. A.; Zinn, B. T.; Stallybrass, M. P.: Boundary integral solutions of three dimensional acoustic radiation problems. J. sound vibration 59, 245-262 (1978) · Zbl 0391.76052
[3] Bernhard, R. J.; Keltie, R. F.: Numerical techniques in acoustic radiation. (1989)
[4] Wu, T. W.; Li, W. L.; Seybert, A. F.: An efficient boundary-element algorithm for multi-frequency acoustical analysis. J. acoust. Soc. am. 94, 447-452 (1993)
[5] Benthien, W.; Schenck, A.: Nonexistence and nonuniqueness problems associated with integral equation methods in acoustics. Comput. & structures 65, 295-305 (1997) · Zbl 0918.76070
[6] Raveendra, S. T.; Vlahopoulos, N.; Glaves, A.: An indirect formulation for multi-valued impedance simulation in structural acoustics. Appl. math. Modelling 22, 379-393 (1998)
[7] Raveendra, S. T.: An efficient indirect boundary element technique for multi-frequency acoustic analysis. Internat. J. Numer. meth. Engrg. 44, 59-76 (1999) · Zbl 0924.76073
[8] Zhang, Z.; Vlahopoulos, N.; Raveendra, S. T.; Allen, T.; Zhang, K. Y.: A computational acoustic field reconstruction process based on an indirect boundary element formulation. J. acoust. Soc. am. 108, 2167-2178 (2000)
[9] Gaul, L.; Wagner, M.; Wentzel, W.; Dumont, N.: Numerical treatment of acoustic problems with the hybrid boundary element method. Internat. J. Solids structures 38, 1871-1888 (2001) · Zbl 1011.76055
[10] COMET/Acoustics: User’s Manual, Version 5.0, Collins & Aikman, Plymouth, MI, 2002
[11] Vavasis, A.: Preconditioning for boundary integral equations. SIAM J. Matrix anal. Appl. 13, 905-925 (1992) · Zbl 0755.65109
[12] Canning, F. X.: Sparse approximation for solving integral equations with oscillatory kernels. SIAM J. Sci. statist. Comput. 13, 71-87 (1992) · Zbl 0749.65093
[13] Atkinson, K. E.; Graham, I. G.: Iterative solution of linear equations arising from the boundary integral method. SIAM J. Sci. statist. Comput. 13, 694-722 (1992) · Zbl 0809.65110
[14] Chen, K.; Amini, S.: Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries. IMA J. Numer. anal. 13, 43-66 (1993) · Zbl 0762.65063
[15] Amini, S.; Maines, N. D.: Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation. Internat. J. Numer. meth. Engrg. 41, 875-898 (1998) · Zbl 0907.65118
[16] Chen, K.; Harris, P. J.: Efficient preconditioners for iterative solution of the boundary element equations for the three-dimensional Helmholtz equation. Appl. numer. Math. 36, 475-489 (2001) · Zbl 0979.65107
[17] Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. comput. 31, 333-390 (1977) · Zbl 0373.65054
[18] Brandt, A.: Multilevel computations: review and recent developments. Lecture notes in pure and applied mathematics 110, 35-62 (1988)
[19] Hackbush, W.; Trottenberg, U.: Multigrid methods II. Proceedings of the second European conference. Lecture notes in mathematics 1228 (1986)
[20] Barnes, J.; Hut, P.: A hierarchical $O(n{\cdot}$logn) force calculation algorithm. Nature 324, 446-449 (1986)
[21] Grama, A.; Kumar, A.; Sameh, A.: Parallel hierarchical solvers and preconditioners for boundary element methods. SIAM J. Sci. comput. 20, 337-358 (1998) · Zbl 0919.65068
[22] Beylkin, G.; Coifman, R.; Rokhlin, V.: Fast wavelet transforms and numerical algorithms. 1. Commun. pure appl. Math. 44, 141-183 (1991) · Zbl 0722.65022
[23] Alpert, B.: A class of bases in L2 for the sparse representation of integral operators. SIAM J. Math. anal. 24, 246-262 (1993) · Zbl 0764.42017
[24] Alpert, B.; Beylkin, G.; Coifman, R.; Rokhlin, V.: Waveletlike bases for the fast solution of second-kind integral equations. SIAM J. Sci. comput. 14, 159-184 (1993) · Zbl 0771.65088
[25] Von Petersdorff, T.; Schwab, C.; Schneider, R.: Multiwavelets for second-kind integral equations. SIAM J. Numer. anal. 34, 2212-2227 (1997) · Zbl 0891.65121
[26] Rokhlin, V.: Rapid solution of integral equations of classical potential theory. J. comput. Phys. 60, 187-207 (1985) · Zbl 0629.65122
[27] Greengard, L.; Rokhlin, V.: A fast algorithm for particle simulations. J. comput. Phys. 73, 325-348 (1987) · Zbl 0629.65005
[28] Rokhlin, V.: Rapid solutions of integral equations of scattering theory in two dimensions. J. comput. Phys. 86, 414-439 (1990) · Zbl 0686.65079
[29] Coifman, R.; Rokhlin, V.; Wandzura, S.: The fast multipole method for the wave equation: a pedestrian prescription. IEEE trans. Antennas and propagation 35, 7-12 (1993)
[30] Lu, C. C.; Chew, W. C.: A multilevel algorithm for solving a boundary integral equation of wave scattering. Microwave opt. Tech. lett. 7, 466-470 (1994)
[31] Nabors, K.; Korsmeyer, F. T.; Leighton, F. T.; White, J.: Preconditioned, adaptive, multipole-accelerated iterative methods for three-dimensional first-kind integral equations of potential theory. SIAM J. Sci. comput. 15, 713-735 (1994) · Zbl 0801.65131
[32] Petersen, H. G.; Smith, E. R.; Soelvason, D.: Error-estimates for the fast multipole method. 2. The 3-dimensional case. Proc. roy. Soc. London A 448, 401-418 (1995) · Zbl 0831.65136
[33] Epton, M. A.; Dembart, B.: Multipole translation theory for the three-dimensional Laplace and Helmholtz equations. SIAM J. Sci. comput. 16, 865-897 (1995) · Zbl 0852.31006
[34] Rahola, J.: Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems. Bit 36, 333-358 (1996) · Zbl 0854.65122
[35] Greengard, L.; Lee, J. Y.: A direct adaptive Poisson solver of arbitrary order accuracy. J. comput. Phys. 125, 415-424 (1996) · Zbl 0851.65090
[36] Cheng, H.; Greengard, L.; Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. comput. Phys. 155, 468-498 (1999) · Zbl 0937.65126
[37] Pham, H. H.; Nathan, A.: A new approach for rapid evaluation of the potential field in three dimensions. Proc. roy. Soc. London A 455, 637-675 (1999) · Zbl 0927.35027
[38] Koc, S.; Song, J.; Chew, W. C.: Error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem. SIAM J. Numer. anal. 36, 906-921 (1999) · Zbl 0924.65116
[39] Darve, E.: The fast multipole method: numerical implementation. J. comput. Phys. 160, 195-240 (2000) · Zbl 0974.78012
[40] Darve, E.: The fast multipole method 1: error analysis and asymptotic complexity. SIAM J. Numer. anal. 38, 98-128 (2000) · Zbl 0974.65033
[41] Gumerov, N. A.; Duraiswami, R.: Computation of scattering from N spheres using multipole reexpansion. J. acoust. Soc. am. 112, 2688-2701 (2002)
[42] Amini, S.; Profit, A. T. J.: Multi-level fast multipole solution of the scattering problem. Engrg. anal. Bound. elem. 27, 547-564 (2003) · Zbl 1039.65084
[43] Hastriter, M. L.; Ohnuki, S.; Chew, W. C.: Error control of the translation operator in 3D MLFMA. Microwave opt. Techn. lett. 37, 184-188 (2003)
[44] Brandt, A.; Lubrecht, A. A.: Multilevel matrix multiplication and fast solution of integral equations. J. comput. Phys. 90, 348-370 (1990) · Zbl 0707.65025
[45] Lubrecht, A.; Ioannides, E.: A fast solution of the dry contact problem and the associated subsurface stress field using mutlilevel techniques. ASME J. Tribol. 113, 128-133 (1991)
[46] Polonsky, I. A.; Keer, L. M.: A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques. Wear 231, 206-219 (1999)
[47] Polonsky, I. A.; Keer, L. M.: Fast methods for solving rough contact problems: a comparative study. ASME J. Tribol. 122, 36-41 (2000)
[48] Venner, C. H.; Lubrecht, A. A.: Multi-level methods in lubrication. (2000) · Zbl 0815.65087
[49] Brandt, A.: Multilevel computations of integral transforms and particle interactions with oscillatory kernels. Comput. phys. Commun. 65, 24-38 (1991) · Zbl 0900.65121
[50] M.M. Grigoriev, G.F. Dargush, A fast multi-level boundary element method for the Laplace equation, Internat. J. Numer. Meth. Engrg., submitted for publication · Zbl 1093.76043
[51] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P.: Numerical recipes in C. The art of scientific computing. (1992) · Zbl 0845.65001
[52] Banerjee, P. K.: The boundary element methods in engineering. (1994)
[53] Hemker, P. W.; Schippers, H.: Multiple grid methods for the solution of Fredholm integral equations of the second kind. Math. comput. 36, 215-232 (1981) · Zbl 0463.65086
[54] Schippers, H.: Multiple grid methods for boundary integral-equations. Numer. math. 46, 351-363 (1985) · Zbl 0543.65015
[55] Von Petersdorff, T.; Stephan, E. P.: On the convergence of the multigrid method for a hypersingular integral equation of the first kind. Numer. math. 57, 379-391 (1990) · Zbl 0702.65102
[56] F.K. Hebeker, On multigrid methods of the first kind for symmetric boundary integral equations of nonnegative order, TH-Darmstadt, Preprint 1120, 1988
[57] Atkinson, K. E.: Two-grid iteration methods for linear integral-equations of the 2nd kind on piecewise-smooth surface in R3. SIAM J. Sci. comput. 15, 1083-1104 (1994) · Zbl 0810.65113
[58] Greenberg, M. D.: Advanced engineering mathematics. (1998) · Zbl 0673.00004
[59] Prudnikov, A. P.; Brychkov, Yu.A.; Marichev, O. I.: Integrals and series. Volume 2: special functions. (1986) · Zbl 0606.33001
[60] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. (1964) · Zbl 0171.38503
[61] Grigoriev, M. M.; Dargush, G. F.: Boundary element methods for transient convective diffusion. Part II: 2d implementation. Comput. methods appl. Mech. engrg. 192, 4313-4335 (2003) · Zbl 1054.76058