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Multigrid methods for the computation of singular solutions and stress intensity factors I:
Corner singularities

Author: Susanne C. Brenner
Journal: Math. Comp. 68 (1999), 559-583
MSC (1991): Primary 65N55, 65N30
MathSciNet review: 1609670
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Abstract: We consider the Poisson equation $-\Delta u=f$ with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain $\Omega $ with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When $f\in L^{2}(\Omega )$, the rate of convergence to the singular solution in the energy norm is shown to be ${\mathcal{O}}(h)$, and the rate of convergence to the stress intensity factors is shown to be ${\mathcal{O}}(h^{1+(\pi /\omega )-\epsilon })$, where $\omega $ is the largest re-entrant angle of the domain and $\epsilon >0$ can be arbitrarily small. The cost of the algorithm is ${\mathcal{O}}(h^{-2})$. When $f\in H^{1}(\Omega )$, the algorithm can be modified so that the convergence rate to the stress intensity factors is ${\mathcal{O}}(h^{2-\epsilon })$. In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be ${\mathcal{O}}(h^{2-\epsilon })$.

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  • 1. I. Babu\v{s}ka, Finite element method for domains with corners, Computing 6 (1970), 264-273. MR 45:2934
  • 2. I. Babu\v{s}ka, R.B. Kellogg and J. Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math. 33 (1979), 447-471. MR 81c:65054
  • 3. I. Babu\v{s}ka and A. Miller, The post-processing approach in the finite element method - part 2: The calculation of stress intensity factors, Int. J. Numer. Methods Engrg. 20 (1984), 1111-1129.
  • 4. R.E. Bank and C.C. Douglas, Sharp estimates for multigrid rates of convergence with general smoothing and acceleration, SIAM J. Numer. Anal. 22 (1985), 617-633. MR 87j:65037
  • 5. R.E. Bank and T.F. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), 35-51. MR 82b:65113
  • 6. M.\v{S}. Birman and G.E. Skvorcov, On the square summability of the highest derivatives of the solution of the Dirichlet problem in a domain with piecewise smooth boundary, Izv. Vyssh. Uchebn. Zaved. Mat. (1962), no. 5, 12-21 (in Russian). MR 26:2731
  • 7. H. Blum and M. Dobrowolski, On finite element methods for elliptic equations on domains with corners, Computing 28 (1982), 53-63. MR 83a:65108
  • 8. M. Bourlard, M. Dauge, M.-S. Lubuma and S. Nicaise, Coefficients of the singularities for elliptic boundary value problems on domains with conical points . Finite element methods on polygonal domains, SIAM Numer. Anal. 29 (1992), 136-155. MR 93a:65146
  • 9. D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the V-cycle, SIAM J. Numer. Anal. 20 (1983), 967-975. MR 85h:65233
  • 10. J.H. Bramble, Multigrid Methods, Longman Scientific & Technical, Essex, 1993. MR 95b:65002
  • 11. J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 113-124. MR 41:7819
  • 12. J.H. Bramble and J.E. Pasciak, The analysis of smoothers for multigrid algorithms, Math. Comp. 58 (1992), 467-488. MR 92f:65146
  • 13. -, New estimates for multilevel algorithms including the V-cycle, Math. Comp. 60 (1993), 447-471. MR 94a:65064
  • 14. -, Uniform convergence estimates for multigrid $V$-cycle algorithms with less than full elliptic regularity, Domain Decomposition Methods in Science and Engineering (Como, 1992), Contemp. Math., vol. 157, Amer. Math. Soc., Providence, RI, 1994, pp. 17-26. MR 95f:65202
  • 15. S.C. Brenner, Overcoming corner singularities by multigrid methods, SIAM J. Numer. Anal (to appear).
  • 16. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York-Berlin-Heidelberg, 1994. MR 95f:65001
  • 17. S.C. Brenner and L.-Y. Sung, Multigrid methods for the computation of singular solutions and stress intensity factors crack singularities, BIT 37 (1997), 623-643. CMP 98:05
  • 18. C. Chen, Optimal points of the stresses for triangular linear elements, Numer. Math. J. Chinese Univ. 2 (1980), 12-20 (in Chinese). MR 83d:65279
  • 19. C. Chen and Y. Huang, High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Publishing House, Changsha, 1995 (in Chinese).
  • 20. P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, New York, Oxford, 1978. MR 58:25001
  • 21. P. Clément, Approximation by finite element functions using local regularization, R.A.I.R.O. R-2 (1975), 77-84. MR 53:4569
  • 22. M. Costabel and M. Dauge, Computation of corner singularities in linear elasticity, Boundary Value Problems and Integral equations in Nonsmooth Domains (M. Costabel and M. Dauge, eds.), Lecture Notes in Pure and Appl. Math., 167, Marcel Dekker, New York, 1995, pp. 59-68. MR 95i:65150
  • 23. M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341, Springer-Verlag, Berlin-Heidelberg, 1988. MR 91a:35078
  • 24. M. Dauge, M.-S. Lubuma and S. Nicaise, Coefficient des singularitiés pour le problème de Dirichlet sur un polygone, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 483-486. MR 88f:35038
  • 25. M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationschrift, Bonn, 1981.
  • 26. G.H. Golub and C.F. Van Loan, Matrix Computations, Second Edition, The Johns Hopkins University Press, Baltimore, 1989. MR 90d:65055
  • 27. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985. MR 86m:35044
  • 28. -, Problèmes aux limites dans les polygones, Mode d'emploi, EDF Bull. Direction Études Rech. Sér. C. Math. Inform. 1 (1986), 21-59. MR 87g:35073
  • 29. -, Singularités en elasticité, Arch. Rat. Mech. Anal. 107 (1989), 157-180. MR 90j:35170
  • 30. -, Singularities in Boundary Value Problems, Masson, Paris, 1992. MR 93h:35004
  • 31. W. Hackbusch, Multi-Grid Methods and Applications, Springer-Verlag, Berlin, 1985. MR 87e:65082
  • 32. V. Kondratiev, Boundary value problems for elliptic equations in domains with conical or angular points, Tran. Moscow Math. Soc. 16 (1967), 227-313. MR 37:1777
  • 33. D. Leguillon and E. Sanchez-Palencia, Computation of Singular Solutions in Elliptic Problems and Elasticity, Masson, Paris, 1987. MR 90m:73015
  • 34. N. Levine, Superconvergent recovery of the gradient from piecewise linear finite-element approximations, IMA J. Numer. Anal. 5 (1985), 407-427. MR 87b:65206
  • 35. Q. Lin and N. Yan, The Construction and Analysis of Highly Effective Finite Elements, Hebei University Press, Baoding, 1996 (in Chinese).
  • 36. J. Mandel, S. McCormick and R. Bank, Variational multigrid theory, Multigrid Methods (S. McCormick, ed.), Frontiers In Applied Mathematics 3, SIAM, Philadelphia, 1987, pp. 131-177. MR 39m:65004
  • 37. S.F. McCormick, Multigrid methods for variational problems: further results, SIAM J. Numer. Anal. 21 (1984), 255-263. MR 85h:65115
  • 38. M. Mu and H. Huang, Extrapolation acceleration and mg methods for calculating stress intensity factors on re-entrant domains, Math. Numer. Sinica 12 (1990), 54-60 (in Chinese). MR 91e:73115
  • 39. S.A. Nazarov and B.A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, Expositions in Mathematics, vol. 13, de Gruyter, Berlin, New York, 1994. MR 95h:35001
  • 40. J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937-958. MR 51:9525
  • 41. L.A. Oganesjan and L.A. Ruhovec, An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary, \v{Z}. Vy\v{c}isl. Mat. i. Mat. Fiz. 9 (1969), 1102-1120; English transl. in USSR Comput. Math. and Math. Phys. 9 (1969). MR 45:4665
  • 42. P. Papadakis, Computational Aspects of the Determination of the Stress Intensity Factors for Two Dimensional Elasticity, Ph.D. Dissertation, University of Maryland, College Park, 1989.
  • 43. P.J. Papadakis and I. Babu\v{s}ka, A numerical procedure for the determination of certain quantities related to the stress intensity factors in two-dimensional elasticity, Comput. Methods Appl. Mech. Engrg. 122 (1995), 69-92. MR 96a:73016
  • 44. R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), 437-445. MR 83e:65180
  • 45. U. Rüde, Mathematical and Computational Techniques for Multilevel Adaptive Methods, Frontiers in Applied Mathematics 13, SIAM, Philadelphia, 1993. MR 95b:65149
  • 46. A.H. Schatz, An analysis of the finite element method for second order elliptic boundary value problems, Mathematical Theory of Finite and Boundary Element Methods, DMV Seminar Band 15, Birkhäuser, Basel, 1990, pp. 9-133. MR 92f:65004
  • 47. R. Scott, Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), 681-697. MR 55:9560
  • 48. L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions, Math. Comp. 54 (1990), 483-493. MR 95j:65021
  • 49. B.A. Szabó and I. Babu\v{s}ka, Computation of the amplitude of stress singular terms for cracks and reentrant corners, Fracture Mechanics: Nineteenth Symposium, ASTM STP 969 (T.A. Cruse, ed.), American Society for Testing and Materials, Philadelphia, 1988, pp. 101-124.
  • 50. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. MR 80i:46032
  • 51. D. Vasilopoulos, On the determination of higher order terms of singular elastic stress fields near corners, Numer. Math. 53 (1988), 51-95. MR 89h:65193
  • 52. L.B. Wahlbin, On the sharpness of certain local estimates for $\mathaccent "017H^{1}$ projections into finite element spaces: Influence of a reentrant corner, Math. Comp. 42 (1984), 1-8. MR 86b:65129
  • 53. -, Local behavior in finite element methods, Handbook of Numerical Analysis, Vol.II (P.G.Ciarlet and J.L. Lions, eds.), Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991, pp. 355-522. MR 92f:65001
  • 54. -, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics 1605, Springer-Verlag, Berlin, 1995. CMP 97:09
  • 55. Z. Yosibash, Numerical Analysis of Singularities and First Derivatives for Elliptic Boundary Value Problems in Two Dimensions, D.Sc. Dissertation, Sever Institute of Technology, Washington University, St. Louis, 1994.
  • 56. Z. Yosibash and B.A. Szabó, Numerical analysis of singularities in two dimensions, Part : Computation of eigenpairs, Int. J. Numer. Methods Engrg. 38 (1996), 2055-2082. MR 96d:73075
  • 57. H. Yserentant, The convergence of multi-level methods for solving finite-element equations in the presence of singularities, Math. Comp. 47 (1986), 399-409. MR 88d:65149
  • 58. S. Zhang, Optimal-order nonnested multigrid methods for solving finite element equations On quasi-uniform meshes, Math. Comp. 55 (1990), 23-36. MR 91g:65268
  • 59. -, Optimal-order nonnested multigrid methods for solving finite element equations On non-quasi-uniform meshes, Math. Comp. 55 (1990), 439-450. MR 91g:65269
  • 60. -, Optimal-order nonnested multigrid methods for solving finite element equations On degenerate meshes, Math. Comp. 64 (1995), 23-49. MR 95c:65194
  • 61. Q.D. Zhu and Q. Lin, The Superconvergence Theory of Finite Elements, Hunan Science and Technology Publishing House, Changsha, 1989 (in Chinese). MR 93j:65191

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Additional Information

Susanne C. Brenner
Affiliation: Department of Mathematics, University of South Carolina, Columbia, SC 29208

Keywords: Multigrid, corner singularities, stress intensity factors, superconver\-gence
Received by editor(s): July 2, 1996
Additional Notes: This work was supported in part by the National Science Foundation under Grant Nos. DMS-94-96275 and DMS-96-00133.
Article copyright: © Copyright 1999 American Mathematical Society

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