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


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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[1] Bernardi, C.: General finite element interpolation on curved domains (in press) · Zbl 0678.65003
[2] Brezzi, F.: On the existence, uniqueness and approximations of saddlepoint problems arising from Lagrangien multipliers. RAIRO Numer. Anal. R.2, 129-151 (1974) · Zbl 0338.90047
[3] Canuto, C., Fujii, H., Quarteroni, A.: Approximation of Symmetry breaking bifurcations for the Rayleigh convection problem (in press) · Zbl 0534.76089
[4] Canuto, C., Maday, Y., Quarteroni, A.: Analysis of the combined finite element and Fourier interpolation. Numer. Math.39, 205-220 (1982) · Zbl 0496.42002
[5] Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput.38, 67-86 (1982) · Zbl 0567.41008
[6] Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[7] Ciarlet, P.G., Raviart, P.A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. In: The mathematical fundations of the finite element method with application to partial differential equations. Aziz, A.K. (ed.) New York: Academic Press, pp. 409-474, 1972 · Zbl 0262.65070
[8] Descloux, J., Rappaz, J.: On numerical approximation of solution branches of nonlinear equations. RAIRO Numer. Anal.16 (4), 319-350 (1982) · Zbl 0505.65016
[9] Girault, V., Raviart, P.A.: Finite element approximation of the Navier-Stokes equations. Lecture Notes in Mathematics, 1979, n0 749. Berlin, Heidelberg, New York: Springer · Zbl 0413.65081
[10] Grisvard, P.: Equations diff?rentielles abstraites. Ann. Sci. Ecole Norm. Sup.4, 311-395 (1969) · Zbl 0193.43502
[11] Lions, J.L., Magenes, F.: Non homogeneous boundary value problems and applications. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0227.35001
[12] Mercier, B., Raugel, G.: R?solution d’un probl?me aux limites dans un ouvert axisym?trique par ?l?ments finis enr, z et s?ries de Fourier en ?. RAIRO Numer. Anal.16 (4), 405-461 (1982) · Zbl 0531.65054
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