Abstract
In [Nkemzi and Jung, 2013] explicit extraction formulas for the computation of the edge flux intensity functions for the Laplacian at axisymmetric edges are presented. The present paper proposes a new adaptation for the Fourier-finite-element method for efficient numerical treatment of boundary value problems for the Poisson equation in axisymmetric domains Ω̂ ⊂ ℝ3 with edges. The novelty of the method is the use of the explicit extraction formulas for the edge flux intensity functions to define a postprocessing procedure of the finite element solutions of the reduced boundary value problems on the two-dimensional meridian of Ω̂. A priori error estimates show that the postprocessing finite element strategy exhibits optimal rate of convergence on regular meshes. Numerical experiments that validate the theoretical results are presented.
Acknowledgment
This work was done while the author B. Nkemzi was visiting the University of Applied Sciences, Dresden, Germany for a short research stay. His visit was supported by the Alexander von Humboldt Foundation, Bonn, Germany in his capacity as an Alexander von Humboldt Alumnus. Boniface Nkemzi is very grateful for the support and also expresses his sincere thanks to the staff of the Faculty of Computer Science and Mathematics for their warm hospitality during his stay. Our sincere thanks go also to the anonymous reviewers who gave us very useful suggestions that led to a significant improvement of the original article.
References
[1] C. Bernardi, M. Dauge, and Y. Maday, Spectral Methods for Axisymmetric Domains. Gauthier-Villars, Paris, 1999.Search in Google Scholar
[2] D. Braess, Finite Elemente. Springer Verlag, Berlin, 1997.10.1007/978-3-662-07233-2Search in Google Scholar
[3] M. Chiarelli and A. Frediani, A computation of the 3-dimensional J-integral for elastic material with a view to application in fracture-mechanics. Engrg. Frac. Mech. 44 (1993), 763–788.10.1016/0013-7944(93)90205-7Search in Google Scholar
[4] Ph. G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.10.1115/1.3424474Search in Google Scholar
[5] P. Ciarlet Jr., B. Jung, S. Kaddouri, S. Labrunie, and J. Zou, The Fourier singular complement method for the Poisson problem. Part I: prismatic domains. Numer. Math. 101 (2005), 423–450.10.1007/s00211-005-0621-6Search in Google Scholar
[6] P. Ciarlet Jr, B. Jung, S. Kaddouri, S. Labrunie, and J. Zou, The Fourier singular complement method for the Poisson problem. Part II: axisymmetric domains. Numer. Math. 102 (2006), 583–610.10.1007/s00211-005-0664-8Search in Google Scholar
[7] P. Clement, Approximation by finite functions using local regularization. RAIRO Modéle Math. Anal. Numér., R-2 (1975), 77–84.10.1051/m2an/197509R200771Search in Google Scholar
[8] M. Costabel, M. Dauge, and Z. Yosibash, A quasi-dual function method for extracting edge stress intensity functions. SIAM J. Math. Anal. 35 (2004), 1177–1202.10.1137/S0036141002404863Search in Google Scholar
[9] M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Lect. Notes in Math., Vol. 1341, Springer-Verlag, Berlin–Heidelberg–New York, 1988.10.1007/BFb0086682Search in Google Scholar
[10] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Advanced Publishing Program, Boston, 1985.Search in Google Scholar
[11] P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris; Springer-Verlag, Berlin, 1992.Search in Google Scholar
[12] B. Heinrich, Singularity functions at axisymmetric edges and their representation by Fourier series. Math. Methods Appl. Sci., 16 (1993), 837–854.10.1002/mma.1670161202Search in Google Scholar
[13] B. Heinrich, The Fourier-finite-element method for Poisson’s equation in axisymmetric domains with edges. SIAM J. Numer. Anal. 33 (1996), 1885–1911.10.1137/S0036142994266108Search in Google Scholar
[14] O. Huber, J. Nickel, and G. Kuhn, On the decomposition of the J-integral for 3D crack problems. Int. J. Fracture64 (1993), 339–348.10.1007/BF00017849Search in Google Scholar
[15] M. Jung and T. Steiden, Das Multigrid-Programmsystem FEMGP zur lösung elliptischer und parabolischer Differentialgleichungen. Version 06.90, Programmdokumentation, Faultät für Mathematik TU Chemnitz, 1990.Search in Google Scholar
[16] Y. P. Kim and J. R. Kweon, The Fourier-finite element method for the Poisson problem on a non-convex polyhedral cylinder. J. Comput. Appl. Math. 233 (2009), 951–968.10.1016/j.cam.2009.08.097Search in Google Scholar
[17] V. A. Kondrat’ev, Singularities of solutions of the Dirichlet problem for an elliptic equation of second order in the neighbourhood of an edge. Differ. Equations13, 1411–1415 (1967).Search in Google Scholar
[18] B. Mercier and G. Raugel, Résolution d’un problèm aux limites dans un ouvert axisymmétrique par éléments finis en r, z et series de Fourier en ϑ. R.A.I.R.O. Analyse numérique16 (1982), 405–461.10.1051/m2an/1982160404051Search in Google Scholar
[19] B. Nkemzi, Numerische Analysis der Fourier-Finite-Elemente-Methode für die Gleichungen der Elastizitätstheorie. PhD Thesis. Tectum Verlag Marburg, 1997.Search in Google Scholar
[20] B. Nkemzi and M. Jung, Flux intensity functions for the Laplacian at axisymmetric edges. Math. Meth. Appl. Sci. 36 (2013), 153–168.10.1002/mma.2578Search in Google Scholar
[21] B. Nkemzi and S. Tanekou, Predictor–corrector p- and hp-finite element method for Poisson’s equation in polygonal domains. Comput. Meth. Appl. Mech. Engrg. 333 (2018), 74–93.10.1016/j.cma.2018.01.027Search in Google Scholar
[22] N. Omer, Z. Yosibash, M. Costabel, and M. Dauge, Edge flux intensity functions in polyhedral domains and their extraction by a quasidual function method. Int. J. Fracture129 (2004), 97–130.10.1023/B:FRAC.0000045717.60837.75Search in Google Scholar
[23] G. Strang and G. J. Fix, An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, N.J., 1973.Search in Google Scholar
[24] B. Weber, Die Fourier-Finite-Elemente-Methode fü elliptische Interfaceprobleme in axisymmetrischen Gebieten. PhD Thesis, Faculty of Mathematics, TU Chemnitz-Zwickau, 1993.Search in Google Scholar
[25] Z. Yosibash, N. Omer, M. Costabel, and M. Dauge, Edge stress intensity functions in polyhedral domains and their extraction by a quasidual function method. Int. J. Fracture136 (2005), 37–73.10.1007/s10704-005-4245-8Search in Google Scholar
[26] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method. Vol. 2: Solid Mechanics. Butterworth-Heinemann, Oxford, 2000.Search in Google Scholar
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