Articles
Estimation of errors and adaptability in Mimetic Differences for a parabolic type problem
Published 2019-07-24
Keywords
- mimetics diffences,
- adaptivity,
- error estimate,
- conservative methods
How to Cite
Calderón, G., Carrillo, J. C., Villamizar, J., Torres, C., & Castillo, J. E. (2019). Estimation of errors and adaptability in Mimetic Differences for a parabolic type problem. Revista UIS Ingenierías, 18(3), 141–150. https://doi.org/10.18273/revuin.v18n3-2019014
Copyright (c) 2019 Revista UIS Ingenierías
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Abstract
In this article, we present un h-adaptive process that defines an optimal mesh to compute the solution of parabolic boundary value problems by using mimetic numerical methods. The estimation of the error, in the spatial variable, is made from the discrete version of the gradient operator. The numerical experiment shows the good behavior of the procedure
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References
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doi:10.1155/2007/12303
[11] J. E. Castillo and G. F. Miranda, Mimetic discretization methods, 5th ed. Chapman and Hall/CRC, 2013.
[12] F. F. Hernández, J. E. Castillo, and G. A. Larrazábal, “Large sparse linear systems arising from mimetic discretization,” Comput. Math. with Appl., vol. 53, no. 1, pp. 1–11, 2007. doi: 10.1016/j.camwa.2006.08.034
[13] C. Bazan, M. Abouali, J. Castillo, and P. Blomgren, “Mimetic finite difference methods in image processing,” Comput. Appl. Math., vol. 30, no. 3, pp. 701–720, 2011. doi: 10.1590/S1807-03022011000300012
[14] M. Abouali and J. Castillo, “Solving Navier-Stokes’ equation using Castillo-Grone’s mimetic difference operators on GPUs,” in 65th Annual Meeting of the APS Division of Fluid Dynamics, 2012, vol. Volume 57, N. 17.
[15] A. Gómez-Polanco, J. M. Guevara-Jordan, and B. Molina, “A mimetic iterative scheme for solving biharmonic equations,” Math. Comput. Model., vol. 57, no. 9–10, pp. 2132–2139, 2013. doi: 10.1016/j.mcm.2011.03.015
[16] J. Blanco, O. Rojas, C. Chacón, J. M. Guevara-Jordan, and J. Castillo, “Tensor formulation of 3-D mimetic finite differences and applications to elliptic problems,” Electron. Trans. Numer. Anal., vol. 45, pp. 457–475, 2016.
[17] E. D. Batista and J. E. Castillo, “Mimetic schemes on non-uniform structured meshes,” Electron. Trans. Numer. Anal., vol. 34, pp. 152–162, 2008.
[18] A. Lugo Jiménez and G. Calderón, Estimación del error y adaptatividad en esquemas miméticos para problemas de contorno, 2nd ed., vol. XXII. Asociación Matemática Venezolana, 2015.
[19] O. C. Zienkiewicz and J. Z. Zhu, “The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique,” Int. J. Numer. Methods Eng., vol. 33, no. 7, pp. 1331–1364, May 1992. doi:
10.1002/nme.1620330702
[20] P. Díez and G. Calderón, “Goal-oriented error estimation for transient parabolic problems,” Comput. Mech., vol. 39, no. 5, pp. 631–646, 2007. doi: 10.1007/s00466-006-0106-1
[2] J. M. Hyman and M. Shashkov, “Approximation of boundary conditions for mimetic finite-difference methods,” Comput. Math. with Appl., vol. 36, no. 5, pp. 79–99, 1998. doi: 10.1016/S0898-1221(98)00152-7
[3] J. Hyman, J. Morel, M. Shashkov, and S. Steinberg, “Mimetic Finite Difference Methods for Diffusion Equations,” Comput. Geosci., vol. 6, no. 3, pp. 333–352, 2002. doi: 10.1023/A:1021282912658
[4] J. Castillo and R. Grone, “A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law,” SIAM J. Matrix Anal. Appl., vol. 25, no. 1, pp. 128–142, Jan. 2003. doi: 10.1137/S0895479801398025
[5] J. E. Castillo and M. Yasuda, “Linear Systems Arising for Second-Order Mimetic Divergence and Gradient Discretizations,” J. Math. Model. Algorithms, vol. 4, no. 1, pp. 67–82, 2005. doi: 10.1007/s10852-004-3523-1
[6] J. M. Guevara-Jordan, S. Rojas, M. Freites-Villegas, and J. E. Castillo, “A New Second Order Finite Difference Conservative Scheme,” Divulg. Matemáticas, vol. 13, no. 1, pp. 35–40, 2005.
[7] F. Solano-Feo, J. M. Guevara-Jordan, O. Rojas, B. Otero, and R. Rodriguez, “A new mimetic scheme for the acoustic wave equation,” J. Comput. Appl. Math., vol. 295, no. 1, pp. 2–12, 2016. doi: 10.1016/j.cam.2015.09.037
[8] G. Sosa Jones, J. Arteaga, and O. Jiménez, “A study of mimetic and finite difference methods for the static diffusion equation,” Comput. Math. with Appl., vol. 76, no. 3, pp. 633–648, 2018. doi: 10.1016/j.camwa.2018.05.004
[9] O. Montilla, C. Cadenas, and J. Castillo, “Matrix approach to mimetic discretizations for differential operators on non-uniform grids,” Math. Comput. Simul., vol. 73, no. 1–4, pp. 215–225, 2006. doi: .matcom.2006.06.025
[10] J. M. Guevara-Jordan, S. Rojas, M. Freites-Villegas, and J. E. Castillo, “Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation,” Adv. Differ. Equations, vol. 2007, no. 1, p. 12303, 2007.
doi:10.1155/2007/12303
[11] J. E. Castillo and G. F. Miranda, Mimetic discretization methods, 5th ed. Chapman and Hall/CRC, 2013.
[12] F. F. Hernández, J. E. Castillo, and G. A. Larrazábal, “Large sparse linear systems arising from mimetic discretization,” Comput. Math. with Appl., vol. 53, no. 1, pp. 1–11, 2007. doi: 10.1016/j.camwa.2006.08.034
[13] C. Bazan, M. Abouali, J. Castillo, and P. Blomgren, “Mimetic finite difference methods in image processing,” Comput. Appl. Math., vol. 30, no. 3, pp. 701–720, 2011. doi: 10.1590/S1807-03022011000300012
[14] M. Abouali and J. Castillo, “Solving Navier-Stokes’ equation using Castillo-Grone’s mimetic difference operators on GPUs,” in 65th Annual Meeting of the APS Division of Fluid Dynamics, 2012, vol. Volume 57, N. 17.
[15] A. Gómez-Polanco, J. M. Guevara-Jordan, and B. Molina, “A mimetic iterative scheme for solving biharmonic equations,” Math. Comput. Model., vol. 57, no. 9–10, pp. 2132–2139, 2013. doi: 10.1016/j.mcm.2011.03.015
[16] J. Blanco, O. Rojas, C. Chacón, J. M. Guevara-Jordan, and J. Castillo, “Tensor formulation of 3-D mimetic finite differences and applications to elliptic problems,” Electron. Trans. Numer. Anal., vol. 45, pp. 457–475, 2016.
[17] E. D. Batista and J. E. Castillo, “Mimetic schemes on non-uniform structured meshes,” Electron. Trans. Numer. Anal., vol. 34, pp. 152–162, 2008.
[18] A. Lugo Jiménez and G. Calderón, Estimación del error y adaptatividad en esquemas miméticos para problemas de contorno, 2nd ed., vol. XXII. Asociación Matemática Venezolana, 2015.
[19] O. C. Zienkiewicz and J. Z. Zhu, “The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique,” Int. J. Numer. Methods Eng., vol. 33, no. 7, pp. 1331–1364, May 1992. doi:
10.1002/nme.1620330702
[20] P. Díez and G. Calderón, “Goal-oriented error estimation for transient parabolic problems,” Comput. Mech., vol. 39, no. 5, pp. 631–646, 2007. doi: 10.1007/s00466-006-0106-1