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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topBerrone, 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

top- I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element method. SIAM J. Numer. Anal.15 (1978) 736–754. Zbl0398.65069
- 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
- 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
- C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math.85 (2000) 579–608. Zbl0962.65096
- 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
- P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978). Zbl0383.65058
- P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.9 (1975) 77–84. Zbl0368.65008
- W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal.33 (1996) 1106–1124. Zbl0854.65090
- 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
- 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
- K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Numer.4 (1995) 105–158. Zbl0829.65122
- 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
- B.P. Lamichhane and B.I. Wohlmuth, Higher order dual Lagrange multiplier spaces for mortar finite element discretizations. Calcolo39 (2002) 219–237. Zbl1168.65414
- 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
- P. Morin, R.H. Nochetto and K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev.44 (2002) 631–658. Zbl1016.65074
- M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math.16 (2002) 47–75. Zbl0997.65123
- M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg.167 (1998) 223–237. Zbl0935.65105
- L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput.54 (1990) 483–493. Zbl0696.65007
- R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley & Sons, Chichester-New York (1996). Zbl0853.65108
- R. Verfürth, A posteriori error estimates for finite element discretization of the heat equations. Calcolo40 (2003) 195–212. Zbl1168.65418
- 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
- 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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