A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems

Alexandre Ern; Sébastien Meunier

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 43, Issue: 2, page 353-375
  • ISSN: 0764-583X

Abstract

top
We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u, is governed by an elliptic equation and the other, say p, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u- and p-components to obtain optimally convergent a priori bounds for all the terms in the error energy norm. Then, a residual-type a posteriori error analysis is performed. Upon extending the ideas of Verfürth for the heat equation [Calcolo40 (2003) 195–212], an optimally convergent bound is derived for the dominant term in the error energy norm and an equivalence result between residual and error is proven. The error bound can be classically split into time error, space error and data oscillation terms. Moreover, upon extending the elliptic reconstruction technique introduced by Makridakis and Nochetto [SIAM J. Numer. Anal.41 (2003) 1585–1594], an optimally convergent bound is derived for the remaining terms in the error energy norm. Numerical results are presented to illustrate the theoretical analysis.

How to cite

top

Ern, Alexandre, and Meunier, Sébastien. "A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems." ESAIM: Mathematical Modelling and Numerical Analysis 43.2 (2008): 353-375. <http://eudml.org/doc/194454>.

@article{Ern2008,
abstract = { We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u, is governed by an elliptic equation and the other, say p, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u- and p-components to obtain optimally convergent a priori bounds for all the terms in the error energy norm. Then, a residual-type a posteriori error analysis is performed. Upon extending the ideas of Verfürth for the heat equation [Calcolo40 (2003) 195–212], an optimally convergent bound is derived for the dominant term in the error energy norm and an equivalence result between residual and error is proven. The error bound can be classically split into time error, space error and data oscillation terms. Moreover, upon extending the elliptic reconstruction technique introduced by Makridakis and Nochetto [SIAM J. Numer. Anal.41 (2003) 1585–1594], an optimally convergent bound is derived for the remaining terms in the error energy norm. Numerical results are presented to illustrate the theoretical analysis. },
author = {Ern, Alexandre, Meunier, Sébastien},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite element method; energy norm; a posteriori error analysis; hydro-mechanical coupling; poroelasticity.; linearly porous medium; backward Euler scheme; conforming finite elements},
language = {eng},
month = {12},
number = {2},
pages = {353-375},
publisher = {EDP Sciences},
title = {A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems},
url = {http://eudml.org/doc/194454},
volume = {43},
year = {2008},
}

TY - JOUR
AU - Ern, Alexandre
AU - Meunier, Sébastien
TI - A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/12//
PB - EDP Sciences
VL - 43
IS - 2
SP - 353
EP - 375
AB - We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u, is governed by an elliptic equation and the other, say p, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u- and p-components to obtain optimally convergent a priori bounds for all the terms in the error energy norm. Then, a residual-type a posteriori error analysis is performed. Upon extending the ideas of Verfürth for the heat equation [Calcolo40 (2003) 195–212], an optimally convergent bound is derived for the dominant term in the error energy norm and an equivalence result between residual and error is proven. The error bound can be classically split into time error, space error and data oscillation terms. Moreover, upon extending the elliptic reconstruction technique introduced by Makridakis and Nochetto [SIAM J. Numer. Anal.41 (2003) 1585–1594], an optimally convergent bound is derived for the remaining terms in the error energy norm. Numerical results are presented to illustrate the theoretical analysis.
LA - eng
KW - Finite element method; energy norm; a posteriori error analysis; hydro-mechanical coupling; poroelasticity.; linearly porous medium; backward Euler scheme; conforming finite elements
UR - http://eudml.org/doc/194454
ER -

References

top
  1. I. Babuška, M. Feistauer and P. Šolín, On one approach to a posteriori error estimates for evolution problems solved by the method-of-lines. Numer. Math.89 (2001) 225–256.  Zbl0993.65103
  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 (2005) 1117–1138.  Zbl1072.65124
  4. M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys.12 (1941) 155–169.  Zbl67.0837.01
  5. C. Chavant and A. Millard, Simulation d'excavation en comportement hydro-mécanique fragile. Technical report, EDF R&D/AMA and CEA/DEN/SEMT (2007) .  URIhttp://www.gdrmomas.org/ex_qualifications.html
  6. Z. Chen and J. Feng, An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comp.73 (2004) 1167–1193.  Zbl1052.65091
  7. P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.9 (1975) 77–84.  Zbl0368.65008
  8. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal.28 (1991) 43–77.  Zbl0732.65093
  9. K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. II. Optimal error estimates in Ɩ∞Ɩ2 and Ɩ∞Ɩ∞. SIAM J. Numer. Anal.32 (1995) 706–740.  Zbl0830.65094
  10. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences159. Springer-Verlag, New York (2004).  Zbl1059.65103
  11. O. Lakkis and Ch. Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comp.75 (2006) 1627–1658.  Zbl1109.65079
  12. Ch. Makridakis and R.H. Nochetto, Ellitpic reconstruction and a posteriori error estimates for elliptic problems. SIAM J. Numer. Anal.41 (2003) 1585–1594.  Zbl1052.65088
  13. S. Meunier, Analyse d'erreur a posteriori pour les couplages hydro-mécaniques et mise en œuvre dans Code_Aster. Ph.D. Thesis, École nationale des ponts et chaussées, France (2007).  
  14. M.A. Murad and A.F.D. Loula, Improved accuracy in finite element analysis of Biot's consolidation problem. Comput. Meth. Appl. Mech. Engrg.95 (1992) 359–382.  Zbl0760.73068
  15. M.A. Murad and A.F.D. Loula, On stability and convergence of finite element approximations of Biot's consolidation problem. Internat. J. Numer. Methods Engrg.37 (1994) 645–667.  Zbl0791.76047
  16. M.A. Murad, V. Thomée and A.F.D. Loula, Asymptotic behavior of semidiscrete finite-element approximations of Biot's consolidation problem. SIAM J. Numer. Anal.33 (1996) 1065–1083.  Zbl0854.76053
  17. M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg.167 (1998) 223–237.  Zbl0935.65105
  18. R.L. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.  Zbl0696.65007
  19. R.E. Showalter, Diffusion in deformable media. IMA Volumes in Mathematics and its Applications131 (2000) 115–130.  
  20. R.E. Showalter, Diffusion in poro-elastic media. J. Math. Anal. Appl.251 (2000) 310–340.  Zbl0979.74018
  21. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997).  Zbl0884.65097
  22. R. Verfürth, A posteriori error estimations and adaptative mesh-refinement techniques. J. Comput. Appl. Math.50 (1994) 67–83.  Zbl0811.65089
  23. R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Chichester, UK (1996).  Zbl0853.65108
  24. R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation. Calcolo40 (2003) 195–212.  Zbl1168.65418
  25. K. von Terzaghi, Theoretical Soil Mechanics. Wiley, New York (1936).  
  26. M. Wheeler, A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal.10 (1973) 723–759.  Zbl0232.35060

NotesEmbed ?

top

You 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.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.