Modelo de solución de flujos incompresibles

A high order term-by-term stabilization solver for incompressible flow problems
T. Chac´n Rebollo1, V. Girault2 o M. G´mez M´rmol1, I. S´nchez Mu˜ oz3 o a a n March 21, 2011

Abstract In this paper we introduce a low-cost high-order stabilized method for the numerical solution of incompressible flow problems. This a projection-stabilized method where single operator terms (in particular, thepressure gradient) are stabilized by specific terms added to the Galerkin formulation. The main innovation introduced is that we replace the projection-stabilized structure by a interpolation-stabilized structure, with reduced computational cost. We prove the stability of our formulation by means of a specific inf-sup condition. We perform a convergence and error estimates analysis, proving the optimalorder of accuracy of our method. We include extensive numerical tests for linear and non-linear flows that fully confirm our theoretical expectations.

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Introduction

This paper deals with the numerical solution of incompressible flows by low-cost and highlyaccurate solvers. This is a challenging issue, as the numerical solution of incompressible flows faces severe stability restrictions.These restrictions essentially concern discretization of the pressure and discretization of dominating convection terms. Eventually, some other dominant terms may be present in the flow equations (e.g., Coriolis forces in geophysical flow equations), that also need a specific treatment in order to be stably discretized. It is well established that these stability restrictions are treated by limiting aconvenient range of high-frequency components of the terms to be stabilized. This is achieved either by enriching the velocity discretization space (Mixed methods), or by adding specific terms to the standard Galerkin discretizations (Stabilized methods). Indeed, this second procedure may be interpreted as an augmented Galerkin method constructed with an enriched velocity space, via bubble finiteelement functions (cf. [9]). However, mixed methods introduce stabilizing degrees of freedom that do not yield accuracy, and thus are more costly than stabilized methods. For this reason, we focus in this paper on stabilized methods.
Departamento de Ecuaciones Diferenciales y An´lisis Num´rico, Apdo. de correos 1160, Universidad de Sevilla, a e 41080 Sevilla, Spain 2 Mathematics Department, Texas A& M University, College Station, Texas, USA, and Laboratoire Jacques-Louis Lions, C.N.R.S & Universit´ Pierre et Marie Curie, Paris 6, 4 Place Jussieu, 75252 Paris Cedex 05, France e 3 Departamento de Matem´tica Aplicada I, Carretera de Utrera Km 1, Universidad de Sevilla, 41013 Sevilla, Spain a
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There exist two classes of stabilized methods: residual-based methods and penaltymethods. The first ones yield high-order discretizations and are strongly consistent, but are more expensive than the second ones as they include more terms in the discrete problem. Therefore, in this paper we follow the second strategy. Stabilized penalty methods were introduced by Brezzi and Pittk¨ranta in [5]. In this work, a to stabilize the pressure discretization, these authors penalize theGalerkin formulation by a pressure laplacian. This method was extended to the term-by-term stabilized method introduced by Chac´n [9]. In this work, a least-squares penalty term is used to stabilize each actual operator o term that needs stabilization. Since these methods are pure penalty methods, their accuracy is limited to first order in the H 1 norm for the velocity and in the L2 norm for thepressure, for both Oseen and Navier-Stokes equations. The projection-stabilization technique designs high order penalty stabilized method. This strategy was introduced by Blasco and Codina in [2]. In this method the Galerkin formulation is enriched with a projection penalty term that acts on the pressure gradient: Only high frequencies of the pressure gradient that are not representable in the velocity...