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We study the best approximation of a divergence-free function by a divergence-free algebraic or trigonometric polynomial and we prove an optimal estimate. In a particular case we give also an optimal result for the polynomial approximation of a function and its divergence.

A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain Ω is partitioned into two subdomains Ω1 and Ω2. Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in Ω. Finally, a hybrid situation in which viscosity is dropped out only in Ω1 is addressed. The latter is motivated by physical applications. In all cases, correct transmission conditions across the interface Γ between Ω1 and Ω2 are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed. The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out. We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size. © 1991 Springer-Verlag

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Summary.We analyse the Spectral Tau method for the approximation of the Stokes system on a square with Dirichlet boundary conditions. We provide an error estimate, in the norm of the Sobolev space H ~, for the approximation of a divergence free vector field with polynomial divergence free vector fields. We apply this result to prove some convergence estimates for the solution of the discrete Stokes problem.Subject Classifications: AMS(MOS): 65N 30; C R : G 1.8. IntroductionIn this paper we study a spectral approximation of the Stokes system in a square with Dirichlet boundary conditions. Several spectral numerical studies are available for the case of nonperiodic Navier-Stokes equations (see [8, 12-15, 20, 22, 23, 25]), theoretical studies can be found in [2] and [19].We study here the Tau method for the Stokes problem with the velocitypressure formulation; for the analysis of this method when periodicity is assumed in some directions we refer to [6,24]. The Tau method (see e.g. [5,10]) consists in searching a discrete solution in the space of polynomials of degree N vanishing on the boundary and projecting the momentum equation on the space of polynomials of degree N -2 . The incompressibility condition is imposed exactly. In order to solve the discrete problem an iterative method was proposed by Haldenwang in [12]; another scheme, based on the influence matrix method, was proposed by Le-Quere and Alziary de Roquefort in [15] for primitive variables and by Vanel et al. in [25] for the stream function formulation. The analysis of these algorithms will be object of further researches.As in the analysis of the Finite Element approximation of the Stokes system the choice of the pressure space is a crucial problem. For the spectral Tau approximation if we choose all the polynomials of degree N as pressure space, the discrete Stokes problem is not well posed In Sect. 2 we prove a stability result for the velocity; the main difficulty is that the discrete velocity belongs to a space which is different from the space in which the momentum equation is projected (the Tau method is of PetrovGalerkin type, we refer e.g. to [,21] for a review of other methods of this type). To prove stability we use a technique which generalises the method introduced by Canuto and Quartroni in [-5] for the analysis of Poisson's equation.In Sect. 3 we prove an "inf-sup condition" for the pressure (see Brezzi [3]) and we deduce existence and uniqueness of the solution for the approximate problem.In Sect. 4 we state an approximation result in Sobolev spaces, in particular we prove an estimate for the approximation of a divergence free vector field by divergence free polynomial vector fields. Finally we prove some convergence estimates, for the solution of the discrete Stokes problem. Notations. Throughout the paper we shall use the Sobolev spaces W s' P(O) and W~'P((2); for p = 2 we shall denote these spaces by H~(Q) and H~(~). Their definitions and properties can be found e.g. in Adams [-1] and in Lions-Magenes [16]. }l'll~,p will denote the ...