Spectral Tau approximation of the two-dimensional stokes problem

Abstract

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 ...

Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials

“…However thanks to the symmetry of the Galerkin procedure, the complexity of the pre-processing stage in the Galerkin case is an order of magnitude less (in terms of N ) than in the tau (and the collocation) case. Furthermore the error estimates of the Galerkin approximation is optimal (see (2.3)) while that of the tau approximation is not (see for instance [20] and [19]). Remark 2.3.…”

“…In the collocation method, we work in the physical space -a set of collocation points; while in the Galerkin and tau methods, we work in the spectral space -the coefficients of the polynomial series. The Galerkin and collocation methods usually lead to optimal error estimates; while the tau method, being a modification of the Galerkin method, leads to non symmetric variational formulations and only sub-optimal error estimates are available (see for instance [19] and [20]). Gottlieb and Orszag in their pioneer book [11] presented an efficient Chebyshev-tau method; on the other hand, they presented a basis for the Galerkin method which leads to full matrices and its application in practice is prohibited.…”

Divergence stability in connection with the $p$-version of the finite element method

“…The estimate in [17], however, concerns the gradient operator ; it is therefore much closer to an (9 (p )-estimate of the &(H~ * ; L 2 )-norm of a right inverse for the « adjoint » divergence operator. We refer the reader to [26] for other estimâtes related to Proposition 1.…”

“…• [17] is somewhat related to the upper bound in Proposition 1. The estimate in [17], however, concerns the gradient operator ; it is therefore much closer to an (9 (p )-estimate of the &(H~ * ; L 2 )-norm of a right inverse for the « adjoint » divergence operator.…”

The Dissipative Spectral Methods for the First Order Linear Hyperbolic Equations

“…The tau method, where the test functions are always taken without any boundary condition constraints, is one of the basic forms of the spectral method [1,14,19,24]. Analysis and the error estimates of the tau method for the equations above have been discussed in [2].…”