In this paper the initial-boundary value problem of the Navier᎐Stokes equations in stream function form is considered. A trilinear form is introduced to deal with the nonlinear term. A weak formulation of this problem is provided. The existence of a weak solution is proved by an auxiliary semi-discrete Faedo᎐Galerkin scheme and a compactness argument. The uniqueness and regularity of the solution are discussed. Finally the convergence of the numerical solution and the converge rate with a certain choice of basis in the Faedo᎐Galerkin method are given. ᮊ 1997 Academic Press
Abstract. The initial-boundary value problem of two-dimensional incompressible fluid flow in stream function form is considered. A prediction-correction Legendre spectral scheme is proposed, which is easy to be performed. The numerical solution possesses the accuracy of second-order in time and higher order in space. The numerical experiments show the high accuracy of this approach.Résumé. Le problème de fluide incompressibleà deux dimensions est considéré sous forme de fonctioncourant. Un schéma de type prédiction-correction spectrale de Legendre est proposé, ce dernierétant facileà mettre en oeuvre. La solution numérique possède une précision de second ordre en temps et d'ordre supérieur en espace. Les résultats numériques montrent la grande préscison de cette approche.
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