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Combined finite element and spectral approximation of the Navier-Stokes equations. (English) Zbl 0614.76021

The authors consider numerical approximation for the stationary Navier- Stokes equation for an incompressible fluid contained in \(Q=\Omega \times I,\quad \Omega\) a bounded domain of \(R^ 2\) and \(I=(0,2\pi)\), with Dirichlet conditions on the boundary \(\Gamma\) of \(\Omega\) and period conditions on \(I=(0,2\pi)\). Let \[ L=\{f^ 0\in H^ 0(Q): \int f=0\}, \]
\[ V=\{v\in H^ 1(Q): v(x,0)=v(x,2\pi),\quad \forall x\in Q;\quad v(0,y)|_{\Gamma}=0\quad \forall y\in I\}^ 3; \] \(H_ p^{\ell,s}(Q)\) be the closure of the space \(C_ p^{\infty}(J,C^{\infty}(\Omega))\) (the \(C^{\infty}\) functions v defined over R, periodic over I) in the norm of \(H^{\ell,s}(Q)=H^ 0(I,H^{\ell}(Q)\cap H^ s(I,H^ 0(Q))\). In V,L one considers the maps H, \[ H(\lambda,u,p)=(f-\lambda \sum^{3}_{i=1}\frac{\partial (u_ iu)}{\partial x_ i},0) \] and T, where T solves the Stokes problem \[ (u,p)=T(g):-\Delta u+\nabla p=g,\quad \nabla u=0. \] Then an equivalent formulation of the stationary Navier-Stokes equation is to seek a solution (\(\lambda\),u,p) of the equation \[ F(\lambda,u,p)=(u,p)+TH(\lambda,u,p)=0. \] The authors consider the numerical approximation of such solutions using a coupling of a finite element scheme over \(\Omega\) and a spectral Fourier approximation in the periodicity direction supposing: There exists a compact interval \(A\subset R^+\) and a continuous function \(\lambda \to y(\lambda)=(u(\lambda),p(\lambda))\) from \(\Lambda\) to \(V\times L\) such that \(F(\lambda,y(\lambda))=0;\quad D_ uF(\lambda,y(\lambda))\) is an isomorphism of \(V\times L\) and a certain a priori estimate on the solutions \[ \sup_{\lambda \in \Lambda}\{\| u(\lambda)\| H_ p^{\rho,\sigma}(Q)+\| p(\lambda)\| H_ p^{\rho -1,\sigma - 1}(Q)\}\leq C \] where \(\rho\) and \(\sigma\) satisfy \(2\rho^{- 1}+\sigma^{-1}<2\). Under such hypotheses they show the existence of approximating solutions \((u_{\delta},p_{\delta})\) defined in discrete spaces \(V_{\delta}\times L_{\delta}\) depending on the mash size h and the largest wave number N, \(h\leq N^{-1}\) such that if \(r\leq \min (\rho,2)\) and if \(f\in H_ p^{\sigma -1,r-1}(Q)\) there is a unique branch \((u_{\delta}(\lambda),p_{\delta}(\lambda))\) of an associated discrete problem such that \[ \sup_{\lambda \in \Lambda}\{\| u(\lambda)-u_{\delta}(\lambda)\|_ v+\| p(\lambda)- p_{\delta}(\lambda)\|_ v\}\leq C(N^{1-\sigma}+h^{r-1}) \] for \(\delta\) sufficiently small.
Reviewer: J.F.Thompson

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:

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