Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 6, pp. 1043-1070.

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.

DOI : https://doi.org/10.1051/m2an:2003005
Classification : 65N25,  65N30,  74G70
Mots clés : quadratic eigenvalue problems, linear elasticity, 3D vertex singularities, finite element methods, error estimates
@article{M2AN_2002__36_6_1043_0,
author = {Apel, Thomas and S\"andig, Anna-Margarete and Solov'ev, Sergey I.},
title = {Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {1043--1070},
publisher = {EDP-Sciences},
volume = {36},
number = {6},
year = {2002},
doi = {10.1051/m2an:2003005},
zbl = {1137.65426},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003005/}
}
Apel, Thomas; Sändig, Anna-Margarete; Solov'ev, Sergey I. Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 6, pp. 1043-1070. doi : 10.1051/m2an:2003005. http://www.numdam.org/articles/10.1051/m2an:2003005/

[1] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. Mckenney and D. Sorensen, LAPACK Users' Guide. SIAM, Philadelphia, PA, third edition (1999). | Zbl 0934.65030

[2] T. Apel, Anisotropic finite elements: Local estimates and applications. Teubner, Stuttgart, Adv. Numer. Math. (1999). Habilitationsschrift. | MR 1716824 | Zbl 0934.65121

[3] T. Apel, V. Mehrmann and D. Watkins, Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures. Comput. Methods Appl. Mech. Engrg. (to appear), Preprint SFB393/01-25, TU Chemnitz (2001). | Zbl 1029.74042

[4] R.E. Barnhill and J.A. Gregory, Interpolation remainder theory from Taylor expansions on triangles. Numer. Math. 25 (1976) 401-408. | Zbl 0304.65075

[5] Z.P. Bažant and L.M. Keer, Singularities of elastic stresses and of harmonic functions at conical notches or inclusions. Internat. J. Solids Structures 10 (1974) 957-964.

[6] A.E. Beagles and A.-M. Sändig, Singularities of rotationally symmetric solutions of boundary value problems for the Lamé equations. ZAMM 71 (1990) 423-431. | Zbl 0751.73009

[7] P. Benner, R. Byers, V. Mehrmann and H. Xu, Numerical computation of deflating subspaces of embedded Hamiltonian pencils. SIAM J. Matrix Anal. Appl. (to appear), Preprint SFB393/99-15, TU Chemnitz (1999).

[8] M. Costabel and M. Dauge, General edge asymptotics of solutions of second order elliptic boundary value problems I, II. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 109-155, 157-184. | Zbl 0791.35033

[9] M. Dauge, Elliptic boundary value problems on corner domains - smoothness and asymptotics of solutions. Lecture Notes in Math. 1341, Springer, Berlin (1988). | Zbl 0668.35001

[10] M. Dauge, Singularities of corner problems and problems of corner singularities, in: Actes du 30ème Congrés d'Analyse Numérique: CANum '98 (Arles, 1998), Soc. Math. Appl. Indust., Paris (1999) 19-40. | Zbl 0921.35048

[11] M. Dauge, “Simple” corner-edge asymptotics. Internet publication, http://www.maths.univ-rennes1.fr/ dauge/publis/corneredge.pdf (2000).

[12] J.W. Demmel, J.R. Gilbert and X.S. Li, SuperLU Users' Guide. Technical Report LBNL-44289, Lawrence Berkeley National Laboratory (1999).

[13] A. Dimitrov, H. Andrä and E. Schnack, Efficient computation of order and mode of corner singularities in 3d-elasticity. Internat. J. Numer. Methods Engrg. 52 (2001) 805-827. | Zbl 1043.74042

[14] A. Dimitrov and E. Schnack, Asymptotical expansion in non-Lipschitzian domains: a numerical approach using $h$-fem. Numer. Linear Algebra Appl. (to appear). | MR 1934872 | Zbl 1071.65550

[15] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston-London-Melbourne, Monographs and Studies in Mathematics 21 (1985). | Zbl 0695.35060

[16] G. Haase, T. Hommel, A. Meyer, and M. Pester, Bibliotheken zur Entwicklung paralleler Algorithmen. Preprint SPC95_20, TU Chemnitz-Zwickau (1995). Updated version of SPC94_4 and SPC93_1.

[17] H. Jeggle and E. Wendland, On the discrete approximation of eigenvalue problems with holomorphic parameter dependence. Proc. Roy. Soc. Edinburgh Sect. A 78 (1977) 1-29. | Zbl 0383.65038

[18] O.O. Karma, Approximation of operator functions and convergence of approximate eigenvalues. Tr. Vychisl. Tsentra Tartu. Gosudarst. Univ. 24 (1971) 3-143. In Russian.

[19] O.O. Karma, Asymptotic error estimates for approximate characteristic value of holomorphic Fredholm operator functions. Zh. Vychisl. Mat. Mat. Fiz. 11 (1971) 559-568. In Russian. | Zbl 0222.47007

[20] O.O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions. I. Numer. Funct. Anal. Optim. 17 (1996) 365-387. | Zbl 0880.47009

[21] O.O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions. II: Convergence rate. Numer. Funct. Anal. Optim. 17 (1996) 389-408. | Zbl 0880.47010

[22] V.A. Kondrat'Ev, Boundary value problems for elliptic equations on domains with conical or angular points. Tr. Mosk. Mat. Obs. 16 (1967) 209-292. In Russian. | Zbl 0162.16301

[23] V.A. Kozlov, V.G. Maz'Ya and J. Roßmann, Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society (1997).

[24] V.A. Kozlov, V.G. Maz'Ya and J. Roßmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. American Mathematical Society (2001). | Zbl 0965.35003

[25] S.G. Krejn and V.P. Trofimov, On holomorphic operator functions of several complex variables. Funct. Anal. Appl. 3 (1969) 85-86. In Russian. English transl. in Funct. Anal. Appl. 3 (1969) 330-331. | Zbl 0201.16901

[26] S.G. Krejn and V.P. Trofimov, On Fredholm operator depending holomorphically on the parameters. Tr. Seminara po funk. anal. Voronezh univ. (1970) 63-85. | Zbl 0276.47029

[27] D. Leguillon, Computation of 3D-singularities in elasticity, in: Boundary value problems and integral equations in nonsmooth domains, M. Costabel, M. Dauge and S. Nicaise Eds. New York, Lecture Notes in Pure and Appl. Math. 167 (1995) 161-170. Marcel Dekker. Proceedings of a conference at CIRM, Luminy, France, May 3-7 (1993). | Zbl 0876.35031

[28] D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity. Masson, Paris (1987). | MR 995254 | Zbl 0647.73010

[29] R.B. Lehoucq, D.C. Sorensen and C. Yang, ARPACK user's guide. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia, PA, Software Environ. Tools 6 (1998). | Zbl 0901.65021

[30] A.S. Markus, On holomorphic operator functions. Dokl. Akad. Nauk 119 (1958) 1099-1102. In Russian. | Zbl 0089.32203

[31] A.S. Markus, Introduction to spectral theory of polynomial operator pencils. American Mathematical Society, Providence (1988). | MR 971506 | Zbl 0678.47005

[32] A.S. Markus and E.I. Sigal, The multiplicity of the characteristic number of an analytic operator function. Mat. Issled. 5 (1970) 129-147. In Russian. | Zbl 0234.47013

[33] V.G. Maz'Ya and B. Plamenevskiĭ, ${L}_{p}$-estimates of solutions of elliptic boundary value problems in domains with edges. Tr. Mosk. Mat. Obs. 37 (1978) 49-93. In Russian. English transl. in Trans. Moscow Math. Soc. 1 (1980) 49-97. | Zbl 0453.35025

[34] V.G. Maz'Ya and B. Plamenevskiĭ, The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries, part I, II. Z. Anal. Anwendungen 2 (1983) 335-359, 523-551. In Russian. | Zbl 0554.35099

[35] V.G. Maz'Ya and J. Roßmann, Über die Asymptotik der Lösung elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138 (1988) 27-53. | Zbl 0672.35020

[36] V.G. Maz'Ya and J. Roßmann, On the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains. Ann. Global Anal. Geom. 9 (1991) 253-303. | Zbl 0753.35013

[37] V.G. Maz'Ya and J. Roßmann, On the behaviour of solutions to the dirichlet problem for second order elliptic equations near edges and polyhedral vertices with critical angles. Z. Anal. Anwendungen 13 (1994) 19-47. | Zbl 0802.35034

[38] V. Mehrmann and D. Watkins, Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/ Hamiltonian pencils. SIAM J. Sci. Comput. 22 (2001) 1905-1925. | Zbl 0986.65033

[39] B. Mercier and G. Raugel, Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en $r,z$ et séries de Fourier en $\theta$. RAIRO Anal. Numér. 16 (1982) 405-461. | Numdam | Zbl 0531.65054

[40] S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundary. Walter de Gruyter, Berlin, Exposition. Math. 13 (1994). | MR 1283387 | Zbl 0806.35001

[41] S. Nicaise, Regularity of the solutions of elliptic systems in polyhedral domains. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 411-429. | Zbl 0918.35031

[42] M. Pester, Grafik-Ausgabe vom Parallelrechner für 2D-Gebiete. Preprint SPC94_24, TU Chemnitz-Zwickau (1994).

[43] G. Raugel, Résolution numérique de problèmes elliptiques dans des domaines avec coins. Ph.D. thesis, Université de Rennes, France (1978).

[44] G. Raugel, Résolution numérique par une méthode d'éléments finis du problème de Dirichlet pour le Laplacien dans un polygone. C. R. Acad. Sci. Paris Sér. I Math. 286 (1978) A791-A794. | Zbl 0377.65058

[45] A.-M. Sändig and R. Sändig, Singularities of non-rotationally symmetric solutions of boundary value problems for the Lamé equations in a three dimensional domain with conical points. Breitenbrunn, Analysis on manifolds with singularities (1990), Teubner-Texte zur Mathematik, Band 131 (1992) 181-193. | Zbl 0820.35039

[46] H. Schmitz, K. Volk and W.L. Wendland, On three-dimensional singularities of elastic fields near vertices. Numer. Methods Partial Differential Equations 9 (1993) 323-337. | Zbl 0771.73014

[47] V. Staroverov, G. Kobelkov, E. Schnack and A. Dimitrov, On numerical methods for flat crack propagation. IMF-Preprint 99-2, Universität Karlsruhe (1999).

[48] V.P. Trofimov, The root subspaces of operators that depend analytically on a parameter. Mat. Issled. 3 (1968) 117-125. In Russian. | Zbl 0234.47010

[49] G.M. Vainikko and O.O. Karma, Convergence rate of approximate methods in an eigenvalue problem with a parameter entering nonlinearly. Zh. Vychisl. Mat. Mat. Fiz. 14 (1974) 1393-1408. In Russian. | Zbl 0339.65030