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

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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

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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

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  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
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  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

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