Abstract
This paper is concerned with a priori error estimates and convergence analysis of the Fourier-finite-element solutions of the Neumann problem for the Lamé equations in axisymmetric domains 
with reentrant edges. The Fourier-FEM combines the approximating Fourier method with respect to the rotational angle using trigonometric polynomials of degree N (N→∞), with the finite element method on the plane meridian domain of 
with mesh size h (h→0) for approximating the Fourier coefficients. The asymptotic behavior of the solution near reentrant edges is described by singularity functions in non-tensor product form and treated numerically by means of finite element method on locally graded meshes. For 
the rate of convergence of the combined approximations in 
is proved to be of the order 
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T. Apel B. Heinrich (1994) ArticleTitleMesh refinement and windowing near edges for some elliptic problem SIAM J. Numer. Anal. 31 695–708 Occurrence Handle10.1137/0731037
T. Apel A.-M. Sändig J. R. Whiteman (1996) ArticleTitleGraded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains Math. Meth. Appl. Sci. 1 63–85 Occurrence Handle10.1002/(SICI)1099-1476(19960110)19:1<63::AID-MMA764>3.0.CO;2-S
J. Argyris H.-P. Mlejnek (1986) Die Methode der finiten Elemente in der elementaren Strukturmechanik, Vol. 1 Vieweg Braunschweig
C. Bernardi M. Dauge Y. Maday (1999) Spectral methods for axisymmetric domains Gauthier-Villars Pris
Buck, K. E.: Zur Berechnung der Verschiebungen und Spannungen in rotationssymmetrischen Körpern unter beliebiger Belastung. Diss. Universität Stuttgart, 1970.
C. Canuto Y. Maday A. Quarteroni (1982) ArticleTitleAnalysis of the combined finite element and Fourier interpolation Numer. Math. 39 205–220 Occurrence Handle10.1007/BF01408694
Ciarlet, P.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978.
P. Clément (1975) ArticleTitleApproximation by finite element functions using local regularization R.A.I.R.O. 9 IssueID2 77–84
Fritzsch, G.: Zur Berechnung des Verschiebungs- und Spannungszustandes von Rotationskörpern unter beliebig verteilter Belastung. Diss. TU Magdeburg, 1979.
P. Grisvard (1985) Elliptic problems in nonsmooth domains Pitman London
P. Grisvard (1989) ArticleTitleSingularités en elasticité Arch. Rat. Mech. Anal. 107 157–180 Occurrence Handle10.1007/BF00286498
Grisvard, P.: Singularities in boundary value problems. Paris: Masson / Berlin: Springer 1992.
Heinrich, B., Lang, B., Weber, B.: Parallel computation of Fourier-finite-element approximations and some experiments. Preprint SPC 95-22, Technische Universität Chemnitz-Zwickau, Fakultät für Mathematik, Mai 1995.
B. Heinrich (1993) ArticleTitleSingularity functions at axisymmetric edges and their representation by Fourier series Math. Meth. Appl. Sci. 16 837–854
B. Heinrich (1996) ArticleTitleThe Fourier finite element method for Poisson's equation in axisymmetric domains with edges SIAM J. Numer. Anal. 33 IssueID5 1885–1911 Occurrence Handle10.1137/S0036142994266108
B. Heinrich B. Weber (1996) ArticleTitleFourier finite-element approximation of elliptic interface problems in axisymmetric domains Math. Meth. Appl. Sci. 19 909–931 Occurrence Handle10.1002/(SICI)1099-1476(19960725)19:11<909::AID-MMA803>3.0.CO;2-U
Heinrich, B.,Nkemzi, B.: The Fourier finite-element method for the Lamé equations in axisymmetric domains. Preprint-Reihe des Chemnitzer SFB 393, 1999.
Křižek, M., Neittaanmäki, P.: Finite element approximation of variational problems and applications. Pitman Monographs and Surveys in Pure and Applied Mathematics 50, 1990.
B. Mercier G. Raugel (1982) ArticleTitleRésolution 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. Anal. Numér. 16 IssueID4 405–461
J. Nečas I. Hlaváček (1981) Mathematical theory of elastic and elasto-plastic bodies: an introduction Elsevier Amsterdam
Nkemzi, B.: Numerische Analysis der Fourier-Finite-Elemente-Methode für die Gleichungen der Elastizitätstheorie. Diss., TU Chemnitz-Zwickau, 1996 (Marburg L : Tectum-Verlag 1997).
B. Nkemzi B. Heinrich (1999) ArticleTitlePartial Fourier approximation of the Lamé equations in axisymmetric domains Math. Meth. Appl. Sci. 22 1017–1041 Occurrence Handle10.1002/(SICI)1099-1476(199908)22:12<1017::AID-MMA68>3.0.CO;2-B
B. Nkemzi (2004) ArticleTitleSigularities in elasticity and their treatment with Fourier series ZAMM 84 IssueID4 252–265 Occurrence Handle10.1002/zamm.200310099
A. Quarteroni A. Valli (1999) Domain decomposition methods for partial differential equations Oxford University Press Oxford
Schultchen, E., Ulonska, H., Wurmnest.: Statische Berechnung von Rotationskörpern unter beliebiger nicht rotationssymmetrischer Belastung mit dem Programmsystem ANTRAS-ROT. Forschungsbericht 35-2, Tech. Mitt. Krupp, 1977.
M. Sedaghat L. R. Hermann (1983) ArticleTitleA nonlinear semi-analytic finite element analysis for nearly axisymmetric solids Comput. Struct. 17 IssueID3 389–401 Occurrence Handle10.1016/0045-7949(83)90131-1
Tilsch, U.: Finite-Element-Methode zur elastostatischen Berechnung rotationssymmetrischer Körper unter Nutzung von Fourieransätzen. Diss. TU Magdeburg, 1994.
O. C. Zienkiewicz (1987) Methode der finiten Elemente Fachbuchverlag Leipzig
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Nkemzi, B. The Fourier Finite-element Approximation of the Lamé Equations in Axisymmetric Domains with Edges. Computing 76, 11–39 (2006). https://doi.org/10.1007/s00607-005-0121-7
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DOI: https://doi.org/10.1007/s00607-005-0121-7