The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

On a variant of the local projection method stable in the SUPG norm

Petr Knobloch

Kybernetika (2009)

  • Volume: 45, Issue: 4, page 634-645
  • ISSN: 0023-5954

Abstract

top
We consider the local projection finite element method for the discretization of a scalar convection-diffusion equation with a divergence-free convection field. We introduce a new fluctuation operator which is defined using an orthogonal L 2 projection with respect to a weighted L 2 inner product. We prove that the bilinear form corresponding to the discrete problem satisfies an inf-sup condition with respect to the SUPG norm and derive an error estimate for the discrete solution.

How to cite

top

Knobloch, Petr. "On a variant of the local projection method stable in the SUPG norm." Kybernetika 45.4 (2009): 634-645. <http://eudml.org/doc/37725>.

@article{Knobloch2009,
abstract = {We consider the local projection finite element method for the discretization of a scalar convection-diffusion equation with a divergence-free convection field. We introduce a new fluctuation operator which is defined using an orthogonal $L^2$ projection with respect to a weighted $L^2$ inner product. We prove that the bilinear form corresponding to the discrete problem satisfies an inf-sup condition with respect to the SUPG norm and derive an error estimate for the discrete solution.},
author = {Knobloch, Petr},
journal = {Kybernetika},
keywords = {finite element method; convection-diffusion equation; stability; inf-sup condition; stabilization; SUPG method; local projection method; error estimates; finite element method; convection-diffusion equation; stability; inf-sup condition; SUPG method; local projection method; error estimates; Dirichlet boundary value problem},
language = {eng},
number = {4},
pages = {634-645},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On a variant of the local projection method stable in the SUPG norm},
url = {http://eudml.org/doc/37725},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Knobloch, Petr
TI - On a variant of the local projection method stable in the SUPG norm
JO - Kybernetika
PY - 2009
PB - Institute of Information Theory and Automation AS CR
VL - 45
IS - 4
SP - 634
EP - 645
AB - We consider the local projection finite element method for the discretization of a scalar convection-diffusion equation with a divergence-free convection field. We introduce a new fluctuation operator which is defined using an orthogonal $L^2$ projection with respect to a weighted $L^2$ inner product. We prove that the bilinear form corresponding to the discrete problem satisfies an inf-sup condition with respect to the SUPG norm and derive an error estimate for the discrete solution.
LA - eng
KW - finite element method; convection-diffusion equation; stability; inf-sup condition; stabilization; SUPG method; local projection method; error estimates; finite element method; convection-diffusion equation; stability; inf-sup condition; SUPG method; local projection method; error estimates; Dirichlet boundary value problem
UR - http://eudml.org/doc/37725
ER -

References

top
  1. A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo 38 (2001), 173–199. MR1890352
  2. A two-level stabilization scheme for the Navier–Stokes equations, In: Numerical Mathematics and Advanced Applications (M. Feistauer et al., eds.), Springer–Verlag, Berlin 2004, pp. 123–130. MR2121360
  3. Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method, SIAM J. Numer. Anal. 43 (2006), 2544–2566. MR2206447
  4. Streamline upwind / Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), 199–259. MR0679322
  5. Basic error estimates for elliptic problems, In: Handbook of Numerical Analysis, Vol. 2 – Finite Element Methods (pt. 1) (P. G. Ciarlet and J. L. Lions, eds.), North-Holland, Amsterdam 1991, pp. 17–351. Zbl0875.65086MR1115237
  6. On the application of local projection methods to convection-diffusion-reaction problems, In: BAIL 2008 – Boundary and Interior Layers (Lecture Notes Comput. Sci. Engrg. 69, A. F. Hegarty, N. Kopteva, E. O’Riordan, and M. Stynes, eds.), Springer–Verlag, Berlin 2009, pp. 183–194. Zbl1180.35051MR2581489
  7. On the stability of finite element discretizations of convection-diffusion-reaction equations, IMA J. Numer. Anal., Advance Access published on August 27, 2009; doi:10.1093/imanum/drp020 
  8. A unified convergence analysis for local projection stabilisations applied to the Oseen problem, M2AN Math. Model. Numer. Anal. 41 (2007), 713–742. MR2362912
  9. Robust Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion-Reaction and Flow Problems. Second edition. Springer–Verlag, Berlin 2008. MR2454024

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.