Abstract
The Fourier method is combined with the Nitsche-finite-element method (as a mortar method) and applied to the Dirichlet problem of the Poisson equation in three-dimensional axisymmetric domains with reentrant edges generating singularities. The approximating Fourier method yields a splitting of the 3D problem into a set of 2D problems on the meridian plane of the given domain. For solving the 2D problems bearing corner singularities, the Nitsche-finite-element method with non-matching meshes and mesh grading near reentrant corners is applied. Using the explicit representation of some singularity function of non-tensor product type, the rate of convergence of the Fourier-Nitsche-mortaring is estimated in some H 1-like norm as well as in the L 2-norm for weak regularity of the solution. Finally, some numerical results are presented.
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References
Arnold D. N., Brezzi F., Cockburn B. and Marini D. (1999). Discontinuous Galerkin methods for elliptic problems. In: Cockburn, B., Karniadakis, G., and Shu, C.-W. (eds) Discontinuous Galerkin methods: Theory, computation and applications, pp 89–101. Springer, Berlin
Becker R., Hansbo P. and Stenberg R. (2003). A finite element method for domain decomposition with non-matching grids. M2AN Math. Model. Numer. Anal. 37(2): 209–225
Belgacem F. B. (1999). The Mortar finite element method with Lagrange multipliers. Numer. Math. 84: 173–197
Bernardi, C., Maday, Y., Patera, A. T.: A new nonconforming approach to domain decomposition: The mortar element method. In: Nonlinear partial differential equations and their applications (Brezis, H. et al., eds.), pp. 13–51. Paris 1994.
Bernardi, C., Dauge, M., Maday, Y.: Spectral methods for axisymmetric domains. Series in Applied Mathematics. Paris: Gauthiers-Villars and Amsterdam: North Holland 1999.
Blum H. (1988). Numerical treatment of corner and crack singularities. In: Stein, E. and Wendland, W. (eds) Finite element and boundary element techniques from mathematical and engineering point of view, pp 172–212. Springer, Wien, New York
Butzer P. and Nessel R. (1971). Fourier analysis and approximation, vol. I. Birkhäuser, Basel
Clément P. (1975). Approximation by finite element functions using local regularization. R.A.I.R.O. 9(2): 77–84
Fritz A., Hüeber S. and Wohlmuth B. (2004). A comparison of Mortar and Nitsche techniques for linear elasticity. Calcolo 41: 115–137
Grisvard P. (1985). Elliptic problems in nonsmooth domains. Pitman, Boston, London, Melbourne
Heinrich B. (1993). Singularity functions at axisymmetric edges and their representation by Fourier series. Math. Meth. Appl. Sci. 16: 837–854
Heinrich B. (1996). The Fourier-finite-element-method for Poisson's equation in axisymmetric domains with edges. SIAM J. Numer. Anal. 33: 1885–1911
Heinrich B. and Pietsch K. (2002). Nitsche type mortaring for some elliptic problem with corner singularities. Computing 68: 217–238
Heinrich B. and Nicaise S. (2003). The Nitsche mortar finite-element method for transmission problems with Singularities. IMA J. Numer. Anal. 23(2): 331–358
Heinrich B. and Jung B. (2006). The Fourier finite-element method with Nitsche mortaring. IMA J. Numer. Anal. 26(3): 541–562
Heinrich, B., Jung, B.: Nitsche- and Fourier-finite-element method for the Poisson equation in axisymmetric domains with reentrant edges. Preprint SFB393/05-16, Technische Universität Chemnitz 2005.
Heinrich B. and Pönitz K. (2005). Nitsche type mortaring for singularly perturbed reaction-diffusion problems. Computing 75(4): 257–279
Jung, M.: On adaptive grids in multilevel methods. In: GAMM-seminar on multigrid-methods (Hengst, S., ed.), pp. 67–80. Berlin: Inst. Angew. Anal. Stoch., Report No. 5, 1993.
Mercier B. and Raugel G. (1982). Résolution d'un problème aux limites dans un ouvert axisymétrique par éléments finis en r, z et séries de Fourier en ϕ. R.A.I.R.O. Anal. Numér. 16(4): 405–461
Nkemzi B. and Heinrich B. (1999). Partial Fourier approximation of the Lamé equations in axisymmetric domains. Math. Meth. Appl. Sci. 22(12): 1017–1041
Oganesjan, L. A., Ruchovec, L. A.: Variacionno-raznostnye metody resenija ellipticeskich uravnenij. Izd. Akad. Nauk Armjanskoj SSR, Erevan, 1979.
Raugel G. (1978). Résolution numérique par une méthode d'éléments finis du probléme Dirichlet pour le Laplacien dans un polygone. C.R. Acad, Sci. Paris, Sér. A 286(18): A791–A794
Stenberg, R.: Mortaring by a method of J.A. Nitsche. In: Computational mechanics. New trends and applications. Barcelona: Centro Int. Métodos Numér. Ing. 1998.
Weber, B.: Die Fourier-Finite-Elemente-Methode für elliptische Interfaceprobleme in axialsymmetrischen Gebieten. PhD thesis, Technische Universität Chemnitz-Zwickau 1994.
Wohlmuth B. I. (2001). Discretization methods and iterative solvers based on domain decomposition. Lecture Notes in Computational Science and Engineering, vol. 17. Springer, Berlin Heidelberg
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Heinrich, B., Jung, B. The Fourier-Nitsche-mortaring for elliptic problems with reentrant edges. Computing 80, 221–246 (2007). https://doi.org/10.1007/s00607-007-0226-2
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DOI: https://doi.org/10.1007/s00607-007-0226-2