Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations

Stefano Berrone

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 3, page 455-484
  • ISSN: 0764-583X

Abstract

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In this paper we derive a posteriori error estimates for the heat equation. The time discretization strategy is based on a θ-method and the mesh used for each time-slab is independent of the mesh used for the previous time-slab. The novelty of this paper is an upper bound for the error caused by the coarsening of the mesh used for computing the solution in the previous time-slab. The technique applied for deriving this upper bound is independent of the problem and can be generalized to other time dependent problems.

How to cite

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Berrone, Stefano. "Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations." ESAIM: Mathematical Modelling and Numerical Analysis 44.3 (2010): 455-484. <http://eudml.org/doc/250784>.

@article{Berrone2010,
abstract = { In this paper we derive a posteriori error estimates for the heat equation. The time discretization strategy is based on a θ-method and the mesh used for each time-slab is independent of the mesh used for the previous time-slab. The novelty of this paper is an upper bound for the error caused by the coarsening of the mesh used for computing the solution in the previous time-slab. The technique applied for deriving this upper bound is independent of the problem and can be generalized to other time dependent problems. },
author = {Berrone, Stefano},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {A posteriori error estimates; transition condition; parabolic problems; a posteriori error estimates; parabolic equations; theta method},
language = {eng},
month = {4},
number = {3},
pages = {455-484},
publisher = {EDP Sciences},
title = {Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations},
url = {http://eudml.org/doc/250784},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Berrone, Stefano
TI - Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/4//
PB - EDP Sciences
VL - 44
IS - 3
SP - 455
EP - 484
AB - In this paper we derive a posteriori error estimates for the heat equation. The time discretization strategy is based on a θ-method and the mesh used for each time-slab is independent of the mesh used for the previous time-slab. The novelty of this paper is an upper bound for the error caused by the coarsening of the mesh used for computing the solution in the previous time-slab. The technique applied for deriving this upper bound is independent of the problem and can be generalized to other time dependent problems.
LA - eng
KW - A posteriori error estimates; transition condition; parabolic problems; a posteriori error estimates; parabolic equations; theta method
UR - http://eudml.org/doc/250784
ER -

References

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  1. I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element method. SIAM J. Numer. Anal.15 (1978) 736–754.  Zbl0398.65069
  2. R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer.10 (2001) 1–102.  Zbl1105.65349
  3. A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp.74 (2004) 1117–1138.  Zbl1072.65124
  4. C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math.85 (2000) 579–608.  Zbl0962.65096
  5. S. Berrone, Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients. ESAIM: M2AM40 (2006) 991–1021.  Zbl1121.65098
  6. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978).  Zbl0383.65058
  7. P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.9 (1975) 77–84.  Zbl0368.65008
  8. W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal.33 (1996) 1106–1124.  Zbl0854.65090
  9. M. Dryja, M.V. Sarkis and O.B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math.72 (1996) 313–348.  Zbl0857.65131
  10. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. V. Long-time integration. SIAM J. Numer. Anal.32 (1995) 1750–1763.  Zbl0835.65117
  11. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Numer.4 (1995) 105–158.  Zbl0829.65122
  12. B.S. Kirk, J.W. Peterson, R. Stogner and S. Petersen, LibMesh. The University of Texas, Austin, CFDLab and Technische Universität Hamburg, Hamburg, .  URIhttp://libmesh.sourceforge.net
  13. B.P. Lamichhane and B.I. Wohlmuth, Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. Calcolo39 (2002) 219–237.  Zbl1168.65414
  14. B.P. Lamichhane, R.P. Stevenson and B.I. Wohlmuth, Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numer. Math.102 (2005) 93–121.  Zbl1082.65120
  15. P. Morin, R.H. Nochetto and K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev.44 (2002) 631–658.  Zbl1016.65074
  16. M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math.16 (2002) 47–75.  Zbl0997.65123
  17. M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg.167 (1998) 223–237.  Zbl0935.65105
  18. L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput.54 (1990) 483–493.  Zbl0696.65007
  19. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley & Sons, Chichester-New York (1996).  Zbl0853.65108
  20. R. Verfürth, A posteriori error estimates for finite element discretization of the heat equations. Calcolo40 (2003) 195–212.  Zbl1168.65418
  21. B.I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal.38 (2000) 989–1012.  Zbl0974.65105
  22. O.C. Zienkiewicz and J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg.24 (1987) 337–357.  Zbl0602.73063

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