## 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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