The two-level local projection stabilization as an enriched one-level approach

The two-level local projection stabilization is considered as a one-level approach in which the enrichments on each element are piecewise polynomial functions. The dimension of the enrichment space can be significantly reduced without losing the convergence order. On triangular meshes, for example, using continuous piecewise polynomials of degree r ≥ 1, only 2r − 1 functions per macro-cell are needed for the enrichment compared to r2 in the two-level approach. In case of the Stokes problem r − 1 functions per macro-cell are already sufficient to guarantee stability and to preserve convergence order. On quadrilateral meshes the corresponding reduction rates are even higher. We give examples of “reduced” two-level approaches and study how the constant in the local inf-sup condition for the one-level and different two-level approaches, respectively, depends on the polynomial degree r.

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

[2]  Almerico Murli,et al.  Numerical Mathematics and Advanced Applications , 2003 .

[3]  Johannes Löwe,et al.  Local Projection Stabilization of Finite Element Methods for Incompressible Flows , 2008 .

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

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

[6]  W. Layton,et al.  A connection between subgrid scale eddy viscosity and mixed methods , 2002, Appl. Math. Comput..

[7]  Lutz Tobiska,et al.  Analysis of a new stabilized higher order finite element method for advection–diffusion equations , 2006 .

[8]  M. Stynes,et al.  Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems , 1996 .

[9]  Johannes Löwe,et al.  Applying Local Projection Stabilization to inf-sup Stable Elements , 2008 .

[10]  Gunar Matthies,et al.  Stabilization of local projection type applied to convection-diffusion problems with mixed boundary conditions. , 2008 .

[11]  Gunar Matthies,et al.  Local projection stabilisation for higher order discretisations of convection-diffusion problems on Shishkin meshes , 2009, Adv. Comput. Math..

[12]  Gunar Matthies,et al.  The Inf-Sup Condition for the Mapped Qk−Pk−1disc Element in Arbitrary Space Dimensions , 2002, Computing.

[13]  Lutz Tobiska.  On the relationship of local projection stabilization to other stabilized methods for one-dimensional advection–diffusion equations , 2009 .

[14]  Roland Becker,et al.  A Two-Level Stabilization Scheme for the Navier-Stokes Equations , 2004 .

[15]  JohnM . Miller,et al.  Robust Computational Techniques for Boundary Layers , 2000 .

[16]  Gunar Matthies,et al.  Local projection stabilisation on S-type meshes for convection–diffusion problems with characteristic layers , 2010, Computing.

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

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

[19]  Malte Braack,et al.  Finite elements with local projection stabilization for incompressible flow problems , 2009 .

[20]  Elwood T. Olsen,et al.  Bounds on spectral condition numbers of matrices arising in the $p$-version of the finite element method , 1995 .

[21]  Giancarlo Sangalli,et al.  Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods , 2007, SIAM J. Numer. Anal..

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

[23]  Malte Braack.  A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes , 2008 .

[24]  L. P. Shishkina,et al.  Difference Methods for Singular Perturbation Problems , 2008 .

[25]  Gunar Matthies,et al.  Local projection methods on layer-adapted meshes for higher order discretisations of convection--diffusion problems , 2009 .

[26]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[27]  Petr Knobloch,et al.  On the stability of finite-element discretizations of convection–diffusion–reaction equations , 2011 .