Abstract. Following a general analysis of convergence for the finite element solution of the stream function formulation of the Navier-Stokes equation in bounded regions of the plane, an algorithm for pressure recovery is presented. This algorithm, which is easy to implement, is then analyzed and conditions ensuring optimality of the approximation are given. An application is made to a standard conforming cubic macroelement.1. Introduction. The purpose of this paper is to provide a formulation and analysis of the stream function approach to the solution of the two-dimensional Navier-Stokes equations in polygonal simply connected domains. As with related approaches , the pressure must be computed separately if it is required, and we give a natural algorithm for this purpose. In addition, we prove that the computed pressure will be "optimal" in the approximation-theoretic sense, for a particular kind of finite element space. Although specific numerical results are not given, they have been obtained by the authors and confirm the theoretical predictions.For notation, let ß be a bounded simply connected domain in R2. L2(ß) is the Hilbert space of square (Lebesgue) integrable functions with norm || • ||0 and Lo(ß) is the subspace of L2(ß) consisting of functions with zero mean. Let Hm(ü) be the usual Sobolev space consisting of functions which together with their (distributional) derivatives up through order m are in L2(ß). Denote the norm on Hm(Q) by || • \\m. Let #0m(ß) be the completion of C0°°(ß) under the || • ||m norm. We equip i/0m(n)

Abstract. Following a general analysis of convergence for the finite element solution of the stream function formulation of the Navier-Stokes equation in bounded regions of the plane, an algorithm for pressure recovery is presented. This algorithm, which is easy to implement, is then analyzed and conditions ensuring optimality of the approximation are given. An application is made to a standard conforming cubic macroelement.1. Introduction. The purpose of this paper is to provide a formulation and analysis of the stream function approach to the solution of the two-dimensional Navier-Stokes equations in polygonal simply connected domains. As with related approaches , the pressure must be computed separately if it is required, and we give a natural algorithm for this purpose. In addition, we prove that the computed pressure will be "optimal" in the approximation-theoretic sense, for a particular kind of finite element space. Although specific numerical results are not given, they have been obtained by the authors and confirm the theoretical predictions.For notation, let ß be a bounded simply connected domain in R2. L2(ß) is the Hilbert space of square (Lebesgue) integrable functions with norm || • ||0 and Lo(ß) is the subspace of L2(ß) consisting of functions with zero mean. Let Hm(ü) be the usual Sobolev space consisting of functions which together with their (distributional) derivatives up through order m are in L2(ß). Denote the norm on Hm(Q) by || • \\m. Let #0m(ß) be the completion of C0°°(ß) under the || • ||m norm. We equip i/0m(n)

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Abstract.Error estimates for the cell discretization algorithm are obtained for polynomial bases used to approximate both Hk(Q) and analytic solutions to selfadjoint elliptic problems. The polynomial implementation of this algorithm can be viewed as a nonconforming version of the h-p finite element method that also can produce the continuous approximations of the h-p method. The examples provided by our experiments provide discontinuous approximations that have errors similar to the finite element results.

Abstract.Error estimates for the cell discretization algorithm are obtained for polynomial bases used to approximate both Hk(Q) and analytic solutions to selfadjoint elliptic problems. The polynomial implementation of this algorithm can be viewed as a nonconforming version of the h-p finite element method that also can produce the continuous approximations of the h-p method. The examples provided by our experiments provide discontinuous approximations that have errors similar to the finite element results.