A Local Error Estimator for the Mimetic Finite Difference Method
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 2, pp. 319-332.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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.
@article{BUMI_2008_9_1_2_a1,
     author = {Beir\~ao da Veiga, L.},
     title = {A {Local} {Error} {Estimator} for the {Mimetic} {Finite} {Difference} {Method}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {319--332},
     publisher = {mathdoc},
     volume = {Ser. 9, 1},
     number = {2},
     year = {2008},
     zbl = {1164.65034},
     mrnumber = {2424296},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a1/}
}
TY  - JOUR
AU  - Beirão da Veiga, L.
TI  - A Local Error Estimator for the Mimetic Finite Difference Method
JO  - Bollettino della Unione matematica italiana
PY  - 2008
SP  - 319
EP  - 332
VL  - 1
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a1/
LA  - en
ID  - BUMI_2008_9_1_2_a1
ER  - 
%0 Journal Article
%A Beirão da Veiga, L.
%T A Local Error Estimator for the Mimetic Finite Difference Method
%J Bollettino della Unione matematica italiana
%D 2008
%P 319-332
%V 1
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a1/
%G en
%F BUMI_2008_9_1_2_a1
Beirão da Veiga, L. A Local Error Estimator for the Mimetic Finite Difference Method. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 2, pp. 319-332. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_2_a1/

[1] M. Ainsworth - J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000). | DOI | MR | Zbl

[2] S. Agmon, Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton, NJ (1965). | MR | Zbl

[3] D. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19 (1982), 742-760. | DOI | MR | Zbl

[4] L. Beirão Da Veiga, A residual based error estimator for the Mimetic Finite Difference method, Numer. Math., 108 (2008), 387-406. | DOI | MR | Zbl

[5] L. Beirão Da Veiga - M. Manzini, 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) | DOI | MR

[6] M. Berndt - K. Lipnikov - J. D. Moulton - M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation. J. Numer. Math., 9 (2001), 253-284. | MR | Zbl

[7] M. Berndt - K. Lipnikov - M. Shashkov - M. F. Wheeler - I. Yotov, Super-convergence of the velocity in mimetic finite difference methods on quadrilaterals. Siam J. Numer. Anal., 43 (2005), 1728-1749. | DOI | MR | Zbl

[8] D. Braess - R. Verfürth, A posteriori error estimators for the Raviart-Thomas element. Siam. J. Numer. Anal., 33 (1996), 2431-2444. | DOI | MR | Zbl

[9] S. C. Brenner - L. R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag (1994). | DOI | MR | Zbl

[10] F. Brezzi - M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | DOI | MR | Zbl

[11] F. Brezzi - K. Lipnikov - M. Shashkov, Convergence of Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes. SIAM J. Num. Anal., 43 (2005), 1872-1896. | DOI | MR | Zbl

[12] F. Brezzi - K. Lipnikov - V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci., 15 (2005), 1533-1553. | DOI | MR | Zbl

[13] F. Brezzi - K. Lipnikov - M. Shashkov, Convergence of Mimetic Finite Difference Methods for Diffusion Problems on Polyhedral Meshes with curved faces. Math. Models Methods Appl. Sci., 16 (2006), 275-298. | DOI | MR | Zbl

[14] F. Brezzi - K. Lipnikov - M. Shashkov - V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes. To appear on Comp. Meth. and Appl. Mech. Engrg. | DOI | MR | Zbl

[15] A. Cangiani - G. Manzini, Flux reconstruction and pressure post-processing in mimetic finite difference methods. Comput. Meth. Appl. Mech. Engrg., 197 (2008), 933-945. | DOI | MR | Zbl

[16] C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. of Comp., 66 (1996), 465-476. | DOI | MR | Zbl

[17] P. G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR | Zbl

[18] G. Girault - P. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag (1986). | DOI | MR | Zbl

[19] J. Hyman - M. Shashkov - M. Steinberg, The numerical solution of diffusion problems in strongly heterogeneus non-isotropic materials. J. Comput. Phys., 132 (1997), 130-148. | DOI | MR | Zbl

[20] Y. Kuznetsov - K. Lipnikov - M. Shashkov, The mimetic finite difference method on polygonal meshes for diffusion-type problems. Comput. Geosci., 8 (2005), 301-324. | DOI | MR | Zbl

[21] K. Lipnikov - J. Morel - M. Shashkov, Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. J. Comput. Phys., 199 (2004), 589-597. | Zbl

[22] K. Lipnikov - M. Shashkov - D. Svyatskiy, The mimetic finite difference discretiza- tion of diffusion problem on unstructured polyhedral meshes. J. Comput. Phys., 211 (2006), 473-491. | DOI | MR | Zbl

[23] R. Liska - M. Shashkov - V. Ganza, Analysis and optimization of inner products for mimetic finite difference methods on triangular grid. Math. and Comp. in Simulation, 67 (2004), 55-66. | DOI | MR | Zbl

[24] C. Lovadina - R. Stenberg, Energy norm a posteriori error estimates for mixed finite element methods. Math. Comp., 75 (2006), 1659-1674. | DOI | MR | Zbl

[25] J. Morel - M. Hall - M. Shaskov, A local support-operators diffusion discretiza- tion scheme for hexahedral meshes. J. of Comput. Phys., 170 (2001), 338-372. | DOI | MR | Zbl

[26] J. Morel - R. Roberts - M. Shashkov, A local support-operators diffusion discretization scheme for quadrilateral r - z meshes. J. of Comput. Phys., 144 (1998), 17-51. | DOI | MR | Zbl

[27] R. Stenberg, Postprocessing schemes for some mixed finite elements. Math. Model. and Numer. Anal., 25 (1991), 151-168. | fulltext EuDML | DOI | MR | Zbl

[28] R. Verfürth, A review of a posteriori error estimation and adaptive mesh refinement. Wiley and Teubner, Stuttgart (1996).