A multiscale stabilization of the streamfunction form of the steady state Navier-Stokes equations

Abstract

Abstract.In this document, the derivation of a multiscale stabilized finite element method for the streamfunction formulation of the two-dimensional, incompressible Navier-Stokes equations is motivated and outlined. A linearized model problem is developed and analyzed through a variational multiscale approach to determine the form of the stabilized terms. IntroductionWhile stabilized finite element methods are in their third decade of development, they continue to be a lively research topic. Starting with the Streamline/Upwind Petrov-Galerkin Method (SUPG) [1,2], whose name reflects the somewhat heuristic construction, the methods evolved and morphed through improved analysis to become the underpinnings of more recent efforts in multiscale analysis and error estimation/indication. In the process, the stabilization became more general as well. The variational multiscale method developed by Hughes et al. [3,4] has provided a general framework for stabilized methods. By recognizing the impact of small scales on larger scales, the multiscale methodology provides a much clearer physical intuition into the mathematics of such technologies.While stabilized finite element methods for second-order problems have been well-established, no stabilized methods for fourth-order problems have been developed as of yet. Fourth-order problems arise in fluid dynamics, for example, by condensing the Navier-Stokes equations and continuity. By introducing a streamfunction for the solenoidal velocity field, one may reduce the steady, two-dimensional Navier-Stokes equations from a set of three second-order partial differential equations to a single fourth-order partial differential equation in terms of a streamfunction. This fourth-order partial differential equation can then be solved using C 1 finite elements.The popularity of streamfunction formulations has been damped by two limitations. First of all, there is an inherent difficulty in extending streamfunction formulations to three dimensions. Second, C 1 finite elements must be employed, which are not trivial to develop or implement. Magnetohydrodynamics (MHD) has caused a renewed interest in the streamfunction form of the Navier-Stokes equations as (1) the geometries in consideration can be decomposed into twodimensional elements in the plane and spectral elements in the toroidal plane and (2) other operators representing MHD physics also benefit from C 1 continuity [5,6].

Rensselaer Component of the Terascale Simulation Tools and Technologies - Final Report

“…Initial efforts looked into extending adaptive discontinuous Galerkin formulations to solve MHD problems [14]. As the PPPL group moved more heavily to the use of high order finite element methods we investigated the potential advantages of stabilized high-order finite element formulations [9]. As the PPPL group began to develop their new high-order code M3D-C1, the RPI TSTT became directly involved with developing the structures needed to support parallel mesh adaptation and to construct the 2-D parallel adaptive mesh control tools needed [2,12].…”