Mixed approximation of eigenvalue problems: A superconvergence result
ESAIM: Mathematical Modelling and Numerical Analysis (2009)
- Volume: 43, Issue: 5, page 853-865
- ISSN: 0764-583X
Access Full Article
top
Abstract
topHow to cite
topGardini, Francesca. "Mixed approximation of eigenvalue problems: A superconvergence result." ESAIM: Mathematical Modelling and Numerical Analysis 43.5 (2009): 853-865. <http://eudml.org/doc/250592>.
@article{Gardini2009,
abstract = {
We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems.
It is known that a similar superconvergence result holds for the mixed approximation
of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized
in a straightforward way to the eigenvalue problem.
Numerical experiments confirm the
superconvergence property and suggest that it also holds for the lowest order
Brezzi-Douglas-Marini approximation.
},
author = {Gardini, Francesca},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Eigenvalue problem; mixed finite element; superconvergence result; eigenvalue problem; mixed finite element method; superconvergence; Laplace problem; Poisson equation; Raviart-Thomas approximation; Neumann boundary condition; Crouzeix-Raviart approximation; numerical experiments},
language = {eng},
month = {4},
number = {5},
pages = {853-865},
publisher = {EDP Sciences},
title = {Mixed approximation of eigenvalue problems: A superconvergence result},
url = {http://eudml.org/doc/250592},
volume = {43},
year = {2009},
}
TY - JOUR
AU - Gardini, Francesca
TI - Mixed approximation of eigenvalue problems: A superconvergence result
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/4//
PB - EDP Sciences
VL - 43
IS - 5
SP - 853
EP - 865
AB -
We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems.
It is known that a similar superconvergence result holds for the mixed approximation
of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized
in a straightforward way to the eigenvalue problem.
Numerical experiments confirm the
superconvergence property and suggest that it also holds for the lowest order
Brezzi-Douglas-Marini approximation.
LA - eng
KW - Eigenvalue problem; mixed finite element; superconvergence result; eigenvalue problem; mixed finite element method; superconvergence; Laplace problem; Poisson equation; Raviart-Thomas approximation; Neumann boundary condition; Crouzeix-Raviart approximation; numerical experiments
UR - http://eudml.org/doc/250592
ER -
References
top- R.A. Adams, Sobolev spaces, Pure and Applied Mathematics65. Academic Press, New York-London (1975).
- A. Alonso, A. Dello Russo and A. Vampa, A posteriori error estimates in finite element acoustic analysis. J. Comput. Appl. Math.117 (2000) 105–119.
- A. Alonso, A. Dello Russo, C. Padra and R. Rodriguez, Accurate pressure post-process of a finite element method for elastoacoustics. Numer. Math.98 (2004) 389–425.
- D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér.19 (1985) 7–32.
- I. Babǔska and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis2, P.G. Ciarlet and J.L. Lions Eds., North Holland (1991).
- D. Boffi, F. Brezzi and L. Gastaldi, On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Sup. Pisa Cl. Sci.25 (1997) 131–154.
- D. Boffi, F. Kikuci and J. Schöberl, Edge element computation of Maxwell's eigenvalues on general quadrilateral meshes. Math. Models Methods Appl. Sci.16 (2006) 265–273.
- J.H. Brandts, Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math.68 (1994) 311–324.
- F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics15. Springer-Verlag, New York (1991).
- P.G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its application4. North Holland, Amsterdam (1978).
- R. Durán, L. Gastaldi and C. Padra, A posteriori error estimations for mixed approximation of eigenvalue problems. Math. Models Methods Appl. Sci.9 (1999) 1165–1178.
- F. Gardini, A posteriori error estimates for eigenvalue problems in mixed form. Ist. lombardo Accd. Sci. Lett. Rend. A.138 (2004) 17–34.
- F. Gardini, A posteriori error estimates for an eigenvalue problem arising from fluid-structure interactions, Computational Fluid and Solid Mechanics. Elsevier, Amsterdam (2005).
- F. Gardini, A posteriori error estimates for eigenvalue problems in mixed form. Ph.D. Thesis, Università degli Studi di Pavia, Pavia, Italy (2005).
- P. Grisvard, Elliptic problem in nonsmooth domains, Monographs and Studies in Mathematics24. Pitman, Boston (1985).
- J-.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Travaux et Recherches Matheḿatiques17. Dunod, Paris (1968).
- L.D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal.22 (1985) 493–496.
Citations in EuDML Documents
topNotesEmbed ?
topYou 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.