Finite element technique for optimal pressure recovery from stream function formulation of viscous flows

Abstract

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)

Isogeometric Analysis and error estimates for high order partial differential equations in fluid dynamics

“…The following result assures the existence and uniqueness of the solution w 2 U of problem (4.11) provided that a smallness condition on the data is satisfied; see for reference [13,32]. The previous result represents a re-elaborated version of the result provided in [13,32], for which we use the following inequality:…”

“…In particular, the operator curl provides an isomorphism between the spaces W g ; W and the following spaces U g :¼ w 2 X : curlwj C D ¼ g n o and U :¼ u 2 X : curluj C D n ¼ 0g, respectively. Therefore, we can recast problem (4.7) in term of the scalar function w 2 U g , called stream function, and it reads: find, for all t 2 ð0; TÞ; wðtÞ 2 U g such that: mð _ wðtÞ;uÞþ aðwðtÞ;uÞþ cðwðtÞ;wðtÞ;uÞ ¼ FðuÞþHðuÞ 8u 2 U; ðaÞ [13,30,32]). We recall that the source term f defining the linear continuous functional FðÁÞ is such that f 2 H 1 ðXÞ and r Á f ¼ 0 in X.…”

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

“…Lemma 4.2 implies that for H small enough, the inf-sup stability condition still holds in X h , i.e., there exists a γ * (ψ) > 0 such that (see for examples in [6,21]…”

“…The one attraction of stream function formation of the NS equations [14] is that the pressure term can be canceled in the weak form, and there is only one scalar unknown to be solved. The representative work in this field is the analysis of convergence for the standard weak formulation by Cayco and Nicolaides [5,6].Two-grid algorithm is a very promising approach for solving the primitive variable formulation (velocity-pressure formulation) of the NS equations [13,18,20]. Moreover, it can be applied to the stream function formulation of the NS equations [10-12, 21, 26].…”

“…The one attraction of stream function formation of the NS equations [14] is that the pressure term can be canceled in the weak form, and there is only one scalar unknown to be solved. The representative work in this field is the analysis of convergence for the standard weak formulation by Cayco and Nicolaides [5,6].…”