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Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en r,z et séries de Fourier en \(\theta\). (French) Zbl 0531.65054

The main goal of this work is to study the approximation of three- dimensional problems with cylindrical symmetry in polar coordinates (r,z). The variational formulation is given in cylindrical coordinates \((r,\theta\),z) in the space \(\tilde H\) with a weight. The function \(\tilde u\in \tilde H\) is expressible as particular Fourier series \(\tilde u(r,\theta,z)=\sum_{n\in {\mathbb{Z}}}u_ n(\eta,z)e^{in\theta}\) where the functions \(u_ n(\eta,z)\) have the form \[ u_ n(\eta,z)=(1/2\pi)\int^{\pi}_{-\pi}\tilde u(r,\theta,z)e^{- in\theta}d\theta. \] Some auxiliary inequalities for the function \(u_ n(\eta,z)\) as well as of its derivatives are given for \(u\in H^ 2(\Omega)\) and \(u\in H^ 3(\Omega)\), respectively. Next some properties of the Sobolev space with a weight are formulated. It is shown that one can use standard finite elements, by assuming only that the family of triangulations used is regular. Finally, the developed theory is applied to problems where only the domain has cylindrical symmetry. The main effort is devoted to numerical procedures, and \(0(N^{1-\ell}+h)\) error bounds for the problem with cylindrical symmetry are proved.
Reviewer: J.Lovíšek

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
74S05 Finite element methods applied to problems in solid mechanics
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References:

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