| Title:
|
On a variant of the local projection method stable in the SUPG norm (English) |
| Author:
|
Knobloch, Petr |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
45 |
| Issue:
|
4 |
| Year:
|
2009 |
| Pages:
|
634-645 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
finite element method |
| Keyword:
|
convection-diffusion equation |
| Keyword:
|
stability |
| Keyword:
|
inf-sup condition |
| Keyword:
|
stabilization |
| Keyword:
|
SUPG method |
| Keyword:
|
local projection method |
| Keyword:
|
error estimates |
| MSC:
|
65N12 |
| MSC:
|
65N15 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 1191.65155 |
| idMR:
|
MR2588629 |
| . |
| Date available:
|
2010-06-02T19:01:11Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140067 |
| . |
| Reference:
|
[1] R. Becker and M. Braack: A finite element pressure gradient stabilization for the Stokes equations based on local projections.Calcolo 38 (2001), 173–199. MR 1890352 |
| Reference:
|
[2] R. Becker and M. Braack: 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. MR 2121360 |
| Reference:
|
[3] M. Braack and E. Burman: Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method.SIAM J. Numer. Anal. 43 (2006), 2544–2566. MR 2206447 |
| Reference:
|
[4] A. N. Brooks and T. J. R. Hughes: 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. MR 0679322 |
| Reference:
|
[5] P. G. Ciarlet: 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. Zbl 0875.65086, MR 1115237 |
| Reference:
|
[6] P. Knobloch: 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. Zbl 1180.35051, MR 2581489 |
| Reference:
|
[7] P. Knobloch and L. Tobiska: 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 |
| Reference:
|
[8] G. Matthies, P. Skrzypacz, and L. Tobiska: A unified convergence analysis for local projection stabilisations applied to the Oseen problem.M2AN Math. Model. Numer. Anal. 41 (2007), 713–742. MR 2362912 |
| Reference:
|
[9] H.-G. Roos, M. Stynes, and L. Tobiska: Robust Numerical Methods for Singularly Perturbed Differential Equations.Convection-Diffusion-Reaction and Flow Problems. Second edition. Springer–Verlag, Berlin 2008. MR 2454024 |
| . |
