Istituto di Matematica Applicata e Tecnologie Informatiche     
Canuto C., Van Kemenade V., Russo A. Stabilized spectral methods for the Navier--Stokes equations: Residual-free bubbles and preconditioning. Preprint ercim.cnr.ian//1997-1044, 1997.
In a series of papers appeared in the last few years, the use of local bubble functions has been advocated as a stabilizing device for curing spurious oscillations in high order spectral Legendre or Chebyshev methods. This work has been motivated by the success of the bubble approach in low order finite element methods, as a general strategy to design stable approximations to saddle point problems such as the Stokes problem, or to convection-dominated problems. It has been shown that adding local bubbles to the usual (piecewise-)polynomial trial and test functions and eliminating them by static condensation leads to a least-square-type approximation of the Stokes problem, and to an SUPG-type approximation of the convection-diffusion problem. Not only the bubble approach provides an elegant unifying framework in which different stabilization techniques may be cast, but it also yields a path to the fully automatic choice of the stabilization parameters. as well as to adaptivity via a-posteriori error estimators. A previous paper by the first two authors deals with the bubble stabilization of Legendre-Galerkin methods for the incompressible Navier-Stokes equations in velocity-pressure formulation. Local bubbles allow the use of equal order polynomials for approximating the velocity and the pressure; they dramatically reduce the oscillations related to poor resolution in the convection-dominated regime; in addition, they enhance the robustness of a continuation method for treating the nonlinearity. All this, without giving up with the formal infinite-order accuracy of a spectral Legendre method. While the paper clearly indicates the feasibility of the bubble approach, it leaves room for improving the practical efficiency of the method, both by fully exploiting the potential of the bubbles in the automatic (user-independent) setting of the stabilization parameters, and by finding appropriate techniques for the solution of the resulting algebraic system of discrete equations. We address these two issues in the present paper. We propose to adapt to spectral methods the recently developed idea of residual-free bubbles, and to resort to a suitably stabilized finite element preconditioner in solving the algebraic system iteratively. The basic idea of the residual-free bubble approach is as follows: we enlarge the space where the solution is sought by auxiliary functions (the ``bubbles'') whose support is localized in a cell (or an element). These functions are determined by the solution of local problems, in such a way that, in the enlarged space, the governing equations are solved exactly within each element. This motivates the name ``residual-free'' bubbles. If we consider a convection-diffusion problem with piecewise constant coefficients and linear finite elements, then the solution of the local problems -- in the convection-dominated regime -- can be obtained essentially without approximation and at a small cost. In more complicated situations, some approximation is needed. This is particularly true for spectral methods, whose basis functions do not have local support; in this case, a strict application of the residual-free bubble approach would lead to global problems to be solved to get the stabilization coefficients. In the previous paper, a projection operator upon piecewise constant functions is introduced to reduce the interaction between bubbles and global polynomials. In the present paper, we shall allow more flexibility in the choice of the local projection operators. As far as the iterative solver is concerned, we investigate a finite element preconditioner for the Stokes equations. Stabilized Q1-Q1 finite elements built on the spectral-collocation grid are used to generate the preconditioner. The strategy for selecting the stabilization parameters is the same as for the spectral scheme. An outline of the paper is as follows. In Section 2 we describe the residual-free bubble stabilization of the spectral element method and in Section 3 we present the numerical results for the regularized driven cavity and for the backward-facing step. Section 4 contains an analysis of the eigenvalues of the discrete Stokes operator and in Section 5 we address the problem of the finite element preconditioning.
Subject 65-XX

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