Two Level Finite Element Technique for Pressure Recovery from Stream Function Formulation of the Navier-Stokes Equations

Abstract

Abstract. We consider two-level finite element discretization methods for the stream function formulation of the Navier-Stokes equations. The two-level method consists of solving a small nonlinear system on the coarse mesh, then solving a linear system on the fine mesh. It is shown in [8] that the errors between the coarse and fine meshes are related superlinearly. This paper presents an algorithm for pressure recovery and a general analysis of convergence for the algorithm. The numerical example for the 2D driven cavity fluid is considered. Streamfunction contours are displayed showing the main features of the flow.

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

“…Our computations show that we get a stable approximation of the unique solution for (12). The Cavity problem is solved using both the streamfunction equation of the Navier-Stokes model (13) and the streamfunction equation of the Ladyzhenskaya model (12). The numerical computations were performed for different choices of the second viscosity parameter ε 1 and different sizes of triangulations.…”

“…We can establish the weak form for the streamfunction equation of the Ladyzhenskaya equation by first multiplying equation (12) by a test function φ ∈ V q , and integrating over the domain Ω and then applying Green's formula twice to get…”

“…We also study the existence and uniqueness of this discretized version. We start by looking at the streamfunction equation of the Navier-Stokes equations which has been studied in [5,6,[11][12][13]26].…”

“…u = v = 0 in all boundaries except y = 1, where u = 1. Cavity flows have been a subject of study for some time [4,12,13,35]. These flows have been widely used as test cases for validating incompressible fluid dynamics algorithms.…”

Comparison of computational times and accuracy for finite element solution of Navier–Stokes equations and Ladyzhenskaya equations

“…The condition derived in [3] which guarantees the uniqueness of the solutions of the stationary LE model is, in some sense, less pessimistic than the analogous condition for the NSE. Also, the analogous condition for the stationary Ladyzhenskaya model [5,2] generally guarantees uniqueness for the higher values of the Reynolds number than that predicted for the Navier-Stokes model. This research also has the advantage of using the streamfunction formulation of LE.…”