Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients

Stefano Berrone

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 40, Issue: 6, page 991-1021
  • ISSN: 0764-583X

Abstract

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In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math.72 (1996) 313–348; Petzoldt, Adv. Comput. Math.16 (2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.

How to cite

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Berrone, Stefano. "Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 40.6 (2007): 991-1021. <http://eudml.org/doc/194348>.

@article{Berrone2007,
abstract = { In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math.72 (1996) 313–348; Petzoldt, Adv. Comput. Math.16 (2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps. },
author = {Berrone, Stefano},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {A posteriori error estimates; parabolic problems; discontinuous coefficients.; a posteriori error estimates; finite elements; discontinuous coefficients; heat equation},
language = {eng},
month = {2},
number = {6},
pages = {991-1021},
publisher = {EDP Sciences},
title = {Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients},
url = {http://eudml.org/doc/194348},
volume = {40},
year = {2007},
}

TY - JOUR
AU - Berrone, Stefano
TI - Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/2//
PB - EDP Sciences
VL - 40
IS - 6
SP - 991
EP - 1021
AB - In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math.72 (1996) 313–348; Petzoldt, Adv. Comput. Math.16 (2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.
LA - eng
KW - A posteriori error estimates; parabolic problems; discontinuous coefficients.; a posteriori error estimates; finite elements; discontinuous coefficients; heat equation
UR - http://eudml.org/doc/194348
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.  
  2. R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Num. (2001) 1–102.  
  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.  
  4. C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math.85 (2000) 579–608.  
  5. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978).  
  6. P. Clément, Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér.9 (1975) 77–84.  
  7. W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal.33 (1996) 1106–1124.  
  8. 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.  
  9. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. V. Long-time integration. SIAM J. Numer. Anal.32 (1995) 1750–1763.  
  10. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Num. (1995) 105–158.  
  11. 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
  12. P. Morin, R.H. Nocetto and K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev.44 (2002) 631–658.  
  13. M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math.16 (2002) 47–75.  
  14. M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg.167 (1998) 223–237.  
  15. R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Ruhr-Universität Bochum, Report 180/1995.  
  16. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley & Sons, Chichester-New York (1996).  
  17. R. Verfürth, A posteriori error estimates for finite element discretization of the heat equations. Calcolo40 (2003) 195–212.  
  18. 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.  

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