A high order term-by-term stabilization solver for incompressible flow problems

In this paper, we introduce a low-cost, high-order stabilized method for the numerical solution of incompressible flow problems. This is a particular type of projection-stabilized method where each targeted operator, such as the pressure gradient or the convection, is stabilized by least-squares terms added to the Galerkin formulation. The main methodological originality is that we replace the projection-stabilized structure by an interpolation-stabilized structure, with reduced computational cost for some choices of the interpolation operator. This stabilization has one level, in the sense that it is defined on a single mesh. We prove the stability of our formulation by means of a specific inf–sup condition, which is the main technical innovation of our paper. We perform a convergence and error estimates analysis, proving the optimal order of accuracy of our method. We include some numerical tests that confirm our theoretical expectations.

[1]  Miguel A. Fernández,et al.  Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence , 2007, Numerische Mathematik.

[2]  L. R. Scott,et al.  A quasi-local interpolation operator¶preserving the discrete divergence , 2003 .

[3]  Tomás Chacón Rebollo.  A term by term stabilization algorithm for finite element solution of incompressible flow problems , 1998 .

[4]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[5]  O. C. Zienkiewicz,et al.  Finite element methods for second order differential equations with significant first derivatives , 1976 .

[6]  Erik Burman,et al.  Interior penalty variational multiscale method for the incompressible Navier-Stokes equation: Monitoring artificial dissipation , 2007 .

[7]  Erik Burman,et al.  Stabilized finite element methods for the generalized Oseen problem , 2007 .

[8]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[9]  R. Codina.  Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods , 2000 .

[10]  Jean-Luc Guermond,et al.  Subgrid stabilized projection method for 2D unsteady flows at high Reynolds numbers , 2006 .

[11]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[12]  J. Guermond.  Stabilization of Galerkin approximations of transport equations by subgrid modelling , 1999 .

[13]  Erik Burman,et al.  Local Projection Stabilization for the Oseen Problem and its Interpretation as a Variational Multiscale Method , 2006, SIAM J. Numer. Anal..

[14]  Christine Bernardi,et al.  Discr'etisations variationnelles de probl`emes aux limites elliptiques , 2004 .

[15]  Ramon Codina,et al.  Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection , 2000 .

[16]  L. Franca,et al.  Stabilized finite element methods. II: The incompressible Navier-Stokes equations , 1992 .

[17]  Gunar Matthies,et al.  Local projection stabilization of equal order interpolation applied to the Stokes problem , 2008, Math. Comput..

[18]  Vivette Girault,et al.  Two-grid finite-element schemes for the transient Navier-Stokes problem , 2001 .

[19]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[20]  Volker John,et al.  On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review , 2007 .

[21]  Petr Knobloch.  A Generalization of the Local Projection Stabilization for Convection-Diffusion-Reaction Equations , 2010, SIAM J. Numer. Anal..

[22]  P. Oswald,et al.  On a BPX-preconditioner for P1 elements , 1993, Computing.

[23]  Gunar Matthies,et al.  A UNIFIED CONVERGENCE ANALYSIS FOR LOCAL PROJECTION STABILISATIONS APPLIED TO THE OSEEN PROBLEM , 2007 .

[24]  C. Bernardi,et al.  Analysis of some finite elements for the Stokes problem , 1985 .

[25]  Béatrice Rivière,et al.  Error analysis for a monolithic discretization of coupled Darcy and Stokes problems , 2014, J. Num. Math..

[26]  Volker John,et al.  On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part II – Analysis for P1 and Q1 finite elements , 2008 .

[27]  Miguel A. Fernández,et al.  Continuous Interior Penalty Finite Element Method for Oseen's Equations , 2006, SIAM J. Numer. Anal..