×

zbMATH — the first resource for mathematics

Trefftz difference schemes on irregular stencils. (English) Zbl 1188.65146
Summary: The recently developed Flexible Local Approximation MEthod (FLAME) [cf. I. Tsukerman, ibid. 211, No. 2, 659–699 (2006; Zbl 1082.65108)] produces accurate difference schemes by replacing the usual Taylor expansion with Trefftz functions – local solutions of the underlying differential equation. This paper advances and casts in a general form a significant modification of FLAME proposed recently by H. Pinheiro and P. Webb [A FLAME Molecule for 3-D electromagnetic scattering, IEEE Trans. Magn. 45, No. 3, 1120–1123 (2009)]: a least-squares fit instead of the exact match of the approximate solution at the stencil nodes. As a consequence of that, FLAME schemes can now be generated on irregular stencils with the number of nodes substantially greater than the number of approximating functions.
The accuracy of the method is preserved but its robustness is improved. For demonstration, the paper presents a number of numerical examples in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering of electromagnetic (acoustic) waves, and wave propagation in a photonic crystal. The examples explore the role of the grid and stencil size, of the number of approximating functions, and of the irregularity of the stencils.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78A30 Electro- and magnetostatics
35Q60 PDEs in connection with optics and electromagnetic theory
78A45 Diffraction, scattering
Software:
FLAME
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Tsukerman, Igor, A class of difference schemes with flexible local approximation, J. comput. phys., 211, 2, 659-699, (2006) · Zbl 1082.65108
[2] Igor Tsukerman, Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves, Springer, 2008.
[3] V.M.A. Leitão, Generalized finite differences using fundamental solutions, Int. J. Numer. Methods Eng., in press. doi: 10.1002/nme.2697.
[4] Tsukerman, Igor, Electromagnetic applications of a new finite-difference calculus, IEEE trans. magn., 41, 7, 2206-2225, (2005)
[5] Tsukerman, Igor; Čajko, František, Photonic band structure computation using FLAME, IEEE trans. magn., 44, 6, 1382-1385, (2008)
[6] Dai, J.; Tsukerman, I.; Rubinstein, A.; Sherman, S., New computational models for electrostatics of macromolecules in solvents, IEEE trans. magn., 43, 4, 1217-1220, (2007)
[7] Tsukerman, Igor, Quasi-homogeneous backward-wave plasmonic structures: theory and accurate simulation, J. opt. A: pure appl. opt., 11, 114025, (2009)
[8] Dai, Jianhua; Tsukerman, Igor, Flexible approximation schemes with adaptive grid refinement, IEEE trans. magn., 44, 6, 1206-1209, (2008)
[9] Pinheiro, H.; Webb, J.P., A FLAME molecule for 3-D electromagnetic scattering, IEEE trans. magn., 45, 3, 1120-1123, (2009)
[10] Nguyen, V.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: A review and computer implementation aspects, Math. comput. simul., 79, 3, 763-813, (2008) · Zbl 1152.74055
[11] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods eng., 37, 2, 229-256, (1994) · Zbl 0796.73077
[12] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. methods appl. mech. eng., 139, 1-4, 3-47, (1996) · Zbl 0891.73075
[13] Krongauz, Y.; Belytschko, T., EFG approximation with discontinuous derivatives, Int. J. numer. methods eng., 41, 7, 1215-1233, (1998) · Zbl 0906.73063
[14] Liu, W.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[15] Babuška, Ivo; Banerjee, Uday; Osborn, John E., Survey of meshless and generalized finite element methods: a unified approach, Acta numer., 12, 1-125, (2003) · Zbl 1048.65105
[16] Sukumar, N., Construction of polygonal interpolants: A maximum entropy epproach, Int. J. numer. methods eng., 61, 12, 2159-2181, (2004) · Zbl 1073.65505
[17] Arroyo, M.; Ortiz, M., Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, Int. J. numer. methods eng., 65, 2167-2202, (2006) · Zbl 1146.74048
[18] Moskow, Shari; Druskin, Vladimir; Habashy, Tarek; Lee, Ping; Davydycheva, Sofia, A finite difference scheme for elliptic equations with rough coefficients using a Cartesian grid nonconforming to interfaces, SIAM J. numer. anal., 36, 2, 442-464, (1999) · Zbl 0926.65105
[19] Harari, Isaac, A survey of finite element methods for time-harmonic acoustics, Comput. methods appl. mech. engrg., 195, 1594-1607, (2006) · Zbl 1122.76056
[20] Lambe, Larry A.; Luczak, Richard; Nehrbass, John W., A new finite difference method for the Helmholtz equation using symbolic computation, Int. J. comput. eng. sci., 4, 1, 121-144, (2003)
[21] Singer, I.; Turkel, E., Sixth-order accurate finite difference schemes for the Helmholtz equation, J. comput. acoust., 14, 3, 339-351, (2006) · Zbl 1198.65210
[22] Baruch, Guy; Fibich, Gadi; Tsynkov, Semyon; Turkel, Eli, Fourth order schemes for time-harmonic wave equations with discontinuous coefficients, Commun. comput. phys., 5, 442-455, (2009) · Zbl 1364.78031
[23] Nabavi, Majid; Kamran Siddiqui, M.H.; Dargahia, Javad, A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation, J. sound vibrat., 307, 3-5, 972-982, (2007)
[24] Sutmann, Godehard, Compact finite difference schemes of sixth order for the Helmholtz equation, J. comput. appl math., 203, 1, 15-31, (2007) · Zbl 1112.65099
[25] Mei, K.K.; Pous, R.; Chen, Z.; Liu, Y.W.; Prouty, M.D., Measured equation of invariance: a new concept in field computation, IEEE trans. antennas propagat., 42, 320-327, (1994)
[26] Mittra, R.; Ramahi, O.M., Absorbing boundary conditions for direct solution of partial differential equations arising in electromagnetic scattering problems, (), 133-173, (Chapter 4)
[27] Boag, A.; Mittra, R., A numerical absorbing boundary condition for finite difference and finite element analysis of open periodic structures, IEEE trans. microwave theory tech., 43, 150-154, (1995)
[28] Boag, A.; Boag, A.; Mittra, R.; Leviatan, Y., A numerical absorbing boundary condition for finite difference and finite element analysis of open structures, Microw. opt. tech. lett., 7, 395-398, (1994)
[29] Ronald Hadley, G., High-accuracy finite-difference equations for dielectric waveguide analysis I: uniform regions and dielectric interfaces, J. lightw. technol., 20, 7, 1210-1218, (2002)
[30] Ronald Hadley, G., High-accuracy finite-difference equations for dielectric waveguide analysis II: dielectric corners, J. lightw. technol., 20, 7, 1219-1231, (2002)
[31] Chiang, Y.-C.; Chiou, Y.-P.; Chang, H.-C., Finite-difference frequency-domain analysis of 2-D photonic crystals with curved dielectric interfaces, J. lightw. technol., 26, 971-976, (2008)
[32] Chiang, Yen-Chung, Higher order finite-difference frequency domain analysis of 2-D photonic crystals with curved dielectric interfaces, Opt. exp., 17, 5, 3305-3315, (2009)
[33] Yuan, J.; Lu, Y., Photonic bandgap calculations with Dirichlet-to-Neumann maps, J. opt. soc. am. A, 23, 3217-3222, (2006)
[34] Hu, Z.; Lu, Y.Y., Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps, Opt. exp., 16, 17383-17399, (2008)
[35] Harrington, Roger F., Time-harmonic electromagnetic fields, (2001), Wiley-IEEE Press
[36] Čajko, František; Tsukerman, Igor, Flexible approximation schemes for wave refraction in negative index materials, IEEE trans. magn., 44, 6, 1378-1381, (2008)
[37] Bykov, V.P., Spontaneous emission in a periodic structure, Sov. phys. JETP (J. exp. theoret. phys.), 35, 2, 269-273, (1972)
[38] Bykov, V.P., Spontaneous emission from a medium with a band spectrum, Sov. J. quant. electron., 4, 7, 861-871, (1975)
[39] Yablonovitch, Eli, Inhibited spontaneous emission in solid-state physics and electronics, Phys. rev. lett., 58, 20, 2059-2062, (1987)
[40] John, Sajeev, Strong localization of photons in certain disordered dielectric superlattices, Phys. rev. lett., 58, 23, 2486-2489, (1987)
[41] Johnson, Steven G.; Joannopoulos, J.D., Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis, Opt. exp., 8, 3, 173-190, (2001)
[42] Sakoda, Kazuaki, Optical properties of photonic crystals, (2005), Springer Berlin, New York
[43] Fujisawa, Takeshi; Koshiba, Masanori, Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits, J. lightw. technol., 22, 2, 684-691, (2004)
[44] Pinheiro, H.; Webb, J.P.; Tsukerman, I., Flexible local approximation models for wave scattering in photonic crystal devices, IEEE trans. magn., 43, 4, 1321-1324, (2007)
[45] Cheng, H.; Greengard, L.; Rokhlin, V., A fast adaptive multipole algorithm in three dimensions, J. comput. phys., 155, 2, 468-498, (1999) · Zbl 0937.65126
[46] Mishchenko, M.I.; Travis, L.D.; Lacis, A.A., Scattering, absorption, and emission of light by small particles, (2002), Cambridge University Press
[47] Jianhua Dai, Helder Pinheiro, J.P. Webb, Igor Tsukerman, Flexible approximation schemes with numerical and semi-analytical bases, submitted for publication. · Zbl 1218.78145
[48] Al Shenk, N., Uniform error estimates for certain narrow Lagrange finite elements, Math. comput., 63, 105-119, (1994) · Zbl 0807.65003
[49] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[50] Tsukerman, Igor; Plaks, Alexander, Refinement strategies and approximation errors for tetrahedral elements, IEEE trans. magn., 35, 3, 1342-1345, (1999)
[51] Tsukerman, Igor; Plaks, Alexander, Comparison of accuracy criteria for approximation of conservative fields on tetrahedra, IEEE trans. magn., 34, 3252-3255, (1998)
[52] Tsukerman, Igor, A general accuracy criterion for finite element approximation, IEEE trans. magn., 34, 2425-2428, (1998)
[53] Tsukerman, Igor, Approximation of conservative fields and the element “edge shape matrix”, IEEE trans. magn., 34, 3248-3251, (1998)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.