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_319_0,
author = {L. Beir\~ao da Veiga},
title = {A Local Error Estimator for the Mimetic Finite Difference Method},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {1},
year = {2008},
pages = {319-332},
zbl = {1164.65034},
mrnumber = {2424296},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2008_9_1_2_319_0}
}
Beirão da Veiga, L. A Local Error Estimator for the Mimetic Finite Difference Method. Bollettino dell'Unione Matematica Italiana, Tome 1 (2008) pp. 319-332. http://gdmltest.u-ga.fr/item/BUMI_2008_9_1_2_319_0/
[1] - , A Posteriori Error Estimation in Finite Element Analysis. Wiley (2000). | MR 1885308 | Zbl 1008.65076
[2] , Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton, NJ (1965). | MR 178246 | Zbl 0142.37401
[3] , An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19 (1982), 742-760. | MR 664882 | Zbl 0482.65060
[4] , A residual based error estimator for the Mimetic Finite Difference method, Numer. Math., 108 (2008), 387-406. | MR 2365823 | Zbl 1144.65067
[5] - , 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) | MR 2468392
[6] - - - , Convergence of mimetic finite difference discretizations of the diffusion equation. J. Numer. Math., 9 (2001), 253-284. | MR 1879474 | Zbl 1014.65114
[7] - - - - , Super-convergence of the velocity in mimetic finite difference methods on quadrilaterals. Siam J. Numer. Anal., 43 (2005), 1728-1749. | MR 2182147 | Zbl 1096.76030
[8] - , A posteriori error estimators for the Raviart-Thomas element. Siam. J. Numer. Anal., 33 (1996), 2431-2444. | MR 1427472 | Zbl 0866.65071
[9] - , The Mathematical Theory of Finite Element Methods. Springer-Verlag (1994). | MR 1278258 | Zbl 0804.65101
[10] - , Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002
[11] - - , Convergence of Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes. SIAM J. Num. Anal., 43 (2005), 1872-1896. | MR 2192322 | Zbl 1108.65102
[12] - - , A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci., 15 (2005), 1533-1553. | MR 2168945 | Zbl 1083.65099
[13] - - , Convergence of Mimetic Finite Difference Methods for Diffusion Problems on Polyhedral Meshes with curved faces. Math. Models Methods Appl. Sci., 16 (2006), 275-298. | MR 2210091 | Zbl 1094.65111
[14] - - - , A new discretization methodology for diffusion problems on generalized polyhedral meshes. To appear on Comp. Meth. and Appl. Mech. Engrg. | MR 2339994 | Zbl 1173.76370
[15] - , Flux reconstruction and pressure post-processing in mimetic finite difference methods. Comput. Meth. Appl. Mech. Engrg., 197 (2008), 933-945. | MR 2376968 | Zbl 1169.76404
[16] , A posteriori error estimate for the mixed finite element method. Math. of Comp., 66 (1996), 465-476. | MR 1408371 | Zbl 0864.65068
[17] , The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR 520174 | Zbl 0383.65058
[18] - , Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag (1986). | MR 851383 | Zbl 0585.65077
[19] - - , The numerical solution of diffusion problems in strongly heterogeneus non-isotropic materials. J. Comput. Phys., 132 (1997), 130-148. | MR 1440338 | Zbl 0881.65093
[20] - - , The mimetic finite difference method on polygonal meshes for diffusion-type problems. Comput. Geosci., 8 (2005), 301-324. | MR 2198170 | Zbl 1088.76046
[21] - - , Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. J. Comput. Phys., 199 (2004), 589-597. | Zbl 1057.65071
[22] - - , The mimetic finite difference discretiza- tion of diffusion problem on unstructured polyhedral meshes. J. Comput. Phys., 211 (2006), 473-491. | MR 2173393 | Zbl 1120.65332
[23] - - , Analysis and optimization of inner products for mimetic finite difference methods on triangular grid. Math. and Comp. in Simulation, 67 (2004), 55-66. | MR 2088898 | Zbl 1058.65115
[24] - , Energy norm a posteriori error estimates for mixed finite element methods. Math. Comp., 75 (2006), 1659-1674. | MR 2240629 | Zbl 1119.65110
[25] - - , A local support-operators diffusion discretiza- tion scheme for hexahedral meshes. J. of Comput. Phys., 170 (2001), 338-372. | MR 1843613 | Zbl 0983.65096
[26] - - , A local support-operators diffusion discretization scheme for quadrilateral r - z meshes. J. of Comput. Phys., 144 (1998), 17-51. | MR 1633033 | Zbl 06917887
[27] , Postprocessing schemes for some mixed finite elements. Math. Model. and Numer. Anal., 25 (1991), 151-168. | MR 1086845 | Zbl 0717.65081
[28] , A review of a posteriori error estimation and adaptive mesh refinement. Wiley and Teubner, Stuttgart (1996).