A Local Error Estimator for the Mimetic Finite Difference Method

L. Beirão da Veiga

Bollettino dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 2, page 319-332
  • ISSN: 0392-4041

Abstract

top
We present a local error indicator for the Mimetic Finite Difference method for diffusion-type problems on polyhedral meshes. We prove the global reliability and local efficiency of the proposed estimator and show its practical performance on a standard test problem.

How to cite

top

Beirão da Veiga, L.. "A Local Error Estimator for the Mimetic Finite Difference Method." Bollettino dell'Unione Matematica Italiana 1.2 (2008): 319-332. <http://eudml.org/doc/290474>.

@article{BeirãodaVeiga2008,
abstract = {We present a local error indicator for the Mimetic Finite Difference method for diffusion-type problems on polyhedral meshes. We prove the global reliability and local efficiency of the proposed estimator and show its practical performance on a standard test problem.},
author = {Beirão da Veiga, L.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {319-332},
publisher = {Unione Matematica Italiana},
title = {A Local Error Estimator for the Mimetic Finite Difference Method},
url = {http://eudml.org/doc/290474},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Beirão da Veiga, L.
TI - A Local Error Estimator for the Mimetic Finite Difference Method
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/6//
PB - Unione Matematica Italiana
VL - 1
IS - 2
SP - 319
EP - 332
AB - We present a local error indicator for the Mimetic Finite Difference method for diffusion-type problems on polyhedral meshes. We prove the global reliability and local efficiency of the proposed estimator and show its practical performance on a standard test problem.
LA - eng
UR - http://eudml.org/doc/290474
ER -

References

top
  1. AINSWORTH, M. - ODEN, J. T., A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000). Zbl1008.65076MR1885308DOI10.1002/9781118032824
  2. AGMON, S., Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton, NJ (1965). Zbl0142.37401MR178246
  3. ARNOLD, D., An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19 (1982), 742-760. Zbl0482.65060MR664882DOI10.1137/0719052
  4. BEIRÃO DA VEIGA, L., A residual based error estimator for the Mimetic Finite Difference method, Numer. Math., 108 (2008), 387-406. Zbl1144.65067MR2365823DOI10.1007/s00211-007-0126-6
  5. BEIRÃO DA VEIGA, L. - MANZINI, M., An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems with general diffusion tensors, preprint IMATI-CNR 17PV07/17/0 (2007) MR2468392DOI10.1002/nme.2377
  6. BERNDT, M. - LIPNIKOV, K. - MOULTON, J. D. - SHASHKOV, M., Convergence of mimetic finite difference discretizations of the diffusion equation. J. Numer. Math., 9 (2001), 253-284. Zbl1014.65114MR1879474
  7. BERNDT, M. - LIPNIKOV, K. - SHASHKOV, M. - WHEELER, M. F. - YOTOV, I., Super-convergence of the velocity in mimetic finite difference methods on quadrilaterals. Siam J. Numer. Anal., 43 (2005), 1728-1749. Zbl1096.76030MR2182147DOI10.1137/040606831
  8. BRAESS, D. - VERFÜRTH, R., A posteriori error estimators for the Raviart-Thomas element. Siam. J. Numer. Anal., 33 (1996), 2431-2444. Zbl0866.65071MR1427472DOI10.1137/S0036142994264079
  9. BRENNER, S. C. - SCOTT, L. R., The Mathematical Theory of Finite Element Methods. Springer-Verlag (1994). Zbl0804.65101MR1278258DOI10.1007/978-1-4757-4338-8
  10. BREZZI, F. - FORTIN, M., Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). Zbl0788.73002MR1115205DOI10.1007/978-1-4612-3172-1
  11. BREZZI, F. - LIPNIKOV, K. - SHASHKOV, M., Convergence of Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes. SIAM J. Num. Anal., 43 (2005), 1872-1896. Zbl1108.65102MR2192322DOI10.1137/040613950
  12. BREZZI, F. - LIPNIKOV, K. - SIMONCINI, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci., 15 (2005), 1533-1553. Zbl1083.65099MR2168945DOI10.1142/S0218202505000832
  13. BREZZI, F. - LIPNIKOV, K. - SHASHKOV, M., Convergence of Mimetic Finite Difference Methods for Diffusion Problems on Polyhedral Meshes with curved faces. Math. Models Methods Appl. Sci., 16 (2006), 275-298. Zbl1094.65111MR2210091DOI10.1142/S0218202506001157
  14. BREZZI, F. - LIPNIKOV, K. - SHASHKOV, M. - SIMONCINI, V., A new discretization methodology for diffusion problems on generalized polyhedral meshes. To appear on Comp. Meth. and Appl. Mech. Engrg. Zbl1173.76370MR2339994DOI10.1016/j.cma.2006.10.028
  15. CANGIANI, A. - MANZINI, G., Flux reconstruction and pressure post-processing in mimetic finite difference methods. Comput. Meth. Appl. Mech. Engrg., 197 (2008), 933-945. Zbl1169.76404MR2376968DOI10.1016/j.cma.2007.09.019
  16. CARSTENSEN, C., A posteriori error estimate for the mixed finite element method. Math. of Comp., 66 (1996), 465-476. Zbl0864.65068MR1408371DOI10.1090/S0025-5718-97-00837-5
  17. CIARLET, P. G., The Finite Element Method for Elliptic Problems. North-Holland (1978). Zbl0383.65058MR520174
  18. GIRAULT, G. - RAVIART, P., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag (1986). Zbl0585.65077MR851383DOI10.1007/978-3-642-61623-5
  19. HYMAN, J. - SHASHKOV, M. - STEINBERG, M., The numerical solution of diffusion problems in strongly heterogeneus non-isotropic materials. J. Comput. Phys., 132 (1997), 130-148. Zbl0881.65093MR1440338DOI10.1006/jcph.1996.5633
  20. KUZNETSOV, Y. - LIPNIKOV, K. - SHASHKOV, M., The mimetic finite difference method on polygonal meshes for diffusion-type problems. Comput. Geosci., 8 (2005), 301-324. Zbl1088.76046MR2198170DOI10.1007/s10596-004-3771-1
  21. LIPNIKOV, K. - MOREL, J. - SHASHKOV, M., Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. J. Comput. Phys., 199 (2004), 589-597. Zbl1057.65071
  22. LIPNIKOV, K. - SHASHKOV, M. - SVYATSKIY, D., The mimetic finite difference discretiza- tion of diffusion problem on unstructured polyhedral meshes. J. Comput. Phys., 211 (2006), 473-491. Zbl1120.65332MR2173393DOI10.1016/j.jcp.2005.05.028
  23. LISKA, R. - SHASHKOV, M. - GANZA, V., Analysis and optimization of inner products for mimetic finite difference methods on triangular grid. Math. and Comp. in Simulation, 67 (2004), 55-66. Zbl1058.65115MR2088898DOI10.1016/j.matcom.2004.05.008
  24. LOVADINA, C. - STENBERG, R., Energy norm a posteriori error estimates for mixed finite element methods. Math. Comp., 75 (2006), 1659-1674. Zbl1119.65110MR2240629DOI10.1090/S0025-5718-06-01872-2
  25. MOREL, J. - HALL, M. - SHASKOV, M., A local support-operators diffusion discretiza- tion scheme for hexahedral meshes. J. of Comput. Phys., 170 (2001), 338-372. Zbl0983.65096MR1843613DOI10.1006/jcph.2001.6736
  26. MOREL, J. - ROBERTS, R. - SHASHKOV, M., A local support-operators diffusion discretization scheme for quadrilateral r - z meshes. J. of Comput. Phys., 144 (1998), 17-51. Zbl06917887MR1633033DOI10.1006/jcph.1998.5981
  27. STENBERG, R., Postprocessing schemes for some mixed finite elements. Math. Model. and Numer. Anal., 25 (1991), 151-168. Zbl0717.65081MR1086845DOI10.1051/m2an/1991250101511
  28. VERFÜRTH, R., A review of a posteriori error estimation and adaptive mesh refinement. Wiley and Teubner, Stuttgart (1996). 

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.