A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp. (English) Zbl 0870.65092
The authors study the properties of the spectrum of the Laplace operator on various 3-dimensional domains of hyperbolic type. After a review of several results quoted from the literature, the authors present their own numerical results on the discrete spectrum of the Laplace operator on some 3-dimensional prisms which are infinite in one direction (this is called the cusp in the article). The finite element method in its advanced forms (higher degree elements, adaptive refinement, etc.) is used as a tool to calculate the eigenvalues. The results are tabulated according to different symmetries of the eigenfunctions. The graphs show a.o. the dependence of the eigenvalues on the parameters of the domains.
Reviewer: P.Burda (Praha)
MSC:
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
Laplace operator; domains of hyperbolic type; discrete spectrum; finite element method; adaptive refinement; eigenvalues; eigenfunctionsReferences:
| [1] | Abramowitz M., Handbook of Mathematical Functions (1965) |
| [2] | Axelsson O., Finite Element Solution of Boundary Value Problems: Theory and Computation (1984) · Zbl 0537.65072 |
| [3] | Babuška I., Accuracy Estimates and Adaptive Refinements in Finite Element Computations pp 3– (1986) |
| [4] | Babuška I., Numerical Mathematics pp 31– (1988) |
| [5] | Babuška I., SIAM J. Numer. Anal. 13 pp 214– (1976) · Zbl 0324.65046 · doi:10.1137/0713021 |
| [6] | Babuška I., SIAM J. Math. Anal. 19 pp 172– (1988) · Zbl 0647.35021 · doi:10.1137/0519014 |
| [7] | Babuška I., Advances in Engineering Software 15 pp 159– (1992) · Zbl 0769.65078 · doi:10.1016/0965-9978(92)90097-Y |
| [8] | Babuška I., SIAM J. Numer. Anal. 26 pp 1534– (1989) · Zbl 0685.65092 · doi:10.1137/0726090 |
| [9] | Babuška I., SIAM J. Numer. Anal. 24 pp 1249– (1987) · Zbl 0701.65042 · doi:10.1137/0724082 |
| [10] | Babuška I., Math. Comp. 52 pp 275– (1989) |
| [11] | Babuška I., Handbook of Numerical Analysis, II: Finite Element Methods pp 641– (1991) |
| [12] | Banerjee U., Numer. Math. 61 pp 145– (1992) · Zbl 0748.65078 · doi:10.1007/BF01385502 |
| [13] | Banerjee U., Numer. Math. 56 pp 735– (1990) · Zbl 0693.65071 · doi:10.1007/BF01405286 |
| [14] | Bank R. E., Adaptive Computational Methods for Partial Differential Equations pp 74– (1983) |
| [15] | Bank R. E., Advances in Computer Methods for Partial Differential Equations III pp 33– (1979) |
| [16] | Bank R. E., Computing 26 pp 91– (1981) · Zbl 0466.65058 · doi:10.1007/BF02241777 |
| [17] | Bank R. E., Scientific computing pp 3– |
| [18] | Bianchi L., Math. Ann. 40 pp 332– (1892) · JFM 24.0188.02 · doi:10.1007/BF01443558 |
| [19] | Bogomolny E., Phys. Rev. Lett. 69 pp 1477– (1992) · Zbl 0968.81514 · doi:10.1103/PhysRevLett.69.1477 |
| [20] | Bolte J., Phys. Rev. Lett. 69 pp 2188– (1992) · Zbl 0968.81515 · doi:10.1103/PhysRevLett.69.2188 |
| [21] | Chatellin F., Spectral Approximation of Linear Operators (1983) |
| [22] | Ciarlet P. G., The Finite Element Method for Elliptic Problems (1978) · Zbl 0383.65058 |
| [23] | Ciarlet P. G., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations pp 409– (1972) · doi:10.1016/B978-0-12-068650-6.50020-4 |
| [24] | Concus P., Sparse Matrix Computations pp 309– (1976) |
| [25] | Elstrodt J., J. reine angew. Math. 360 pp 160– (1985) |
| [26] | Elstrodt J., Zapiski Nauchnykh Seminarov Mat. Inst. Steklov. 162 pp 77– (1987) |
| [27] | Elstrodt J., ”Groups acting on 3-dimensional hyperbolic space” |
| [28] | Ergatoudis J. G., Internat. J. Solids Structures 4 pp 31– (1968) · Zbl 0152.42802 · doi:10.1016/0020-7683(68)90031-0 |
| [29] | Ewing R. E., Computer Methods in Applied Mechanics and Engineering 82 pp 59– (1990) · Zbl 0731.73095 · doi:10.1016/0045-7825(90)90158-I |
| [30] | Gambolati G., Internat. J. Numer. Methods Engrg. 37 pp 605– (1994) · Zbl 0796.65047 · doi:10.1002/nme.1620370405 |
| [31] | Grunewald F., ”A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with two cusps” · Zbl 0870.65092 |
| [32] | Hackbusch W., Theorie und Numerik elliptischer Differentialgleichungen (1986) · Zbl 0609.65065 · doi:10.1007/978-3-322-99946-7 |
| [33] | Hackbusch W., Iterative Lösung grof{\(\beta\)}r schwachbesetzter Gleichungssysteme,, 2. ed. (1993) |
| [34] | Hejhal D. A., The Selberg Trace Formula for PSL(2,R) 2 (1983) · Zbl 0543.10020 |
| [35] | Hejhal D. A., Mem. Amer. Math. Soc. 97 pp 165– (1992) |
| [36] | Hejhal D. A., Experimental Math. 1 pp 275– (1992) · Zbl 0813.11035 · doi:10.1080/10586458.1992.10504562 |
| [37] | Hestenes M. R., J. Res. Nat. Bur. Standards 49 pp 409– (1952) · Zbl 0048.09901 · doi:10.6028/jres.049.044 |
| [38] | Hughes T. J. R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987) · Zbl 0634.73056 |
| [39] | Huntebrinker W., Bonner Math. Schriften 225 (1991) |
| [40] | Huntebrinker W., Dissertation, in: ”Numerische Bestimmung von Eigenwerten des Laplace-Beltrami-Operators auf dreidimensionalen hyperbolischen Räumen mit Finite-Element-Methoden” (1995) · Zbl 0835.65124 |
| [41] | Jamet P., RAIRO Anal. Numér. 10 pp 43– (1976) |
| [42] | Krizek M., SIAM Journal on Numerical Analysis 29 pp 513– (1992) · Zbl 0755.41003 · doi:10.1137/0729031 |
| [43] | Maa{\(\beta\)} H., Math. Ann. 121 pp 141– (1949) · doi:10.1007/BF01329622 |
| [44] | Maa{\(\beta\)} H., Abh. Math. Sem. Univ. Hamburg 16 pp 72– (1949) |
| [45] | Phillips R. S., Invent. Math. 80 pp 339– (1985) · Zbl 0558.10017 · doi:10.1007/BF01388610 |
| [46] | Phillips R. S., Comm. Pure Appl. Math. 38 pp 853– (1985) · Zbl 0614.10027 · doi:10.1002/cpa.3160380614 |
| [47] | Picard E., Bull. Soc. Math. France 12 pp 43– (1884) |
| [48] | Sarnak P., The Selberg Trace Formula and Related Topics pp 393– (1986) · doi:10.1090/conm/053/853570 |
| [49] | Sarnak P., Festschrift in honour of I. I. PiatetskiShapiro pp 237– (1990) |
| [50] | Sartoletto F., J. Comput. Phys. 81 pp 53– (1989) · Zbl 0664.65033 · doi:10.1016/0021-9991(89)90064-8 |
| [51] | Schwarz H. R., Methode der finiten Elemente,, 3. ed. (1991) · doi:10.1007/978-3-663-10784-2 |
| [52] | Selberg A., J. Indian Math. Soc. (New Series) 20 pp 47– (1956) |
| [53] | Smotrov M. N., ”Small eigenvalues of the Laplacian on T/H3for T= PSL2(Z[i])” (1991) |
| [54] | Steil G., ”Eigenvalues of the Laplacian and of the Hecke operators for PSL(2,Z)” · Zbl 0982.11028 |
| [55] | Stramm K., Schriftenreihe des Mathematischen Instituts und des Graduiertenkollegs der Universitat Miinster, 3. Serie 11 (1994) |
| [56] | Strouboulis T., Comput. Methods Appl. Mech. Engrg. 97 pp 399– (1992) · Zbl 0764.65064 · doi:10.1016/0045-7825(92)90053-M |
| [57] | Swan R. G., Adv. Math. 6 pp 1– (1971) · Zbl 0221.20060 · doi:10.1016/0001-8708(71)90027-2 |
| [58] | Venkov A. B., Dokl. Akad. Nauk SSSR 233 pp 1021– (1977) |
| [59] | Venkov A. B., Dokl. Akad. Nauk SSSR 236 pp 525– (1977) |
| [60] | Venkov A. B., Dokl. Akad. Nauk SSSR 239 pp 511– (1978) |
| [61] | Venkov A. B., Dokl. Akad. Nauk SSSR 247 pp 540– (1979) |
| [62] | Venkov A. B., Trudy Mat. Inst. Steklov. 153 pp 172– (1981) |
| [63] | Watkins D. S., SIAM Review 35 pp 430– (1993) · Zbl 0786.65032 · doi:10.1137/1035090 |
| [64] | Wolpert S. A., Ann. Math. 139 pp 239– (1994) · Zbl 0826.11024 · doi:10.2307/2946582 |
| [65] | Zienkiewicz O. C., Finite Elements and Approximation (1983) · Zbl 0582.65068 |
| [66] | Zienkiewicz O. C., Internat. J. Numer. Methods Engrg. 32 pp 783– (1991) · Zbl 0755.65119 · doi:10.1002/nme.1620320409 |
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