Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T07:31:49.825Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete Maximum Principle and a Delaunay-Type Mesh Condition for Linear Finite Element Approximations of Two-Dimensional Anisotropic Diffusion Problems

Published online by Cambridge University Press:  28 May 2015

Weizhang Huang
Weizhang Huang*
Affiliation:
Department of Mathematics, the University of Kansas, Lawrence, KS 66045, USA
*
*Corresponding author.Email address:huang@math.ku.edu
Get access

Abstract

A Delaunay-type mesh condition is developed for a linear finite element approximation of two-dimensional anisotropic diffusion problems to satisfy a discrete maximum principle. The condition is weaker than the existing anisotropic non-obtuse angle condition and reduces to the well known Delaunay condition for the special case with the identity diffusion matrix. Numerical results are presented to verify the theoretical findings.

Keywords

65N3065N50Anisotropic diffusiondiscrete maximum principlefinite elementmesh generationDelaunay triangulationDelaunay condition
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 3 , August 2011 , pp. 319 - 334
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aavatsmark, I., Barkve, T., Bøe, Ø., and Mannseth, T.. Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAMJ. Sci. Comput, 19:1700–1716 (electronic), 1998.CrossRefGoogle Scholar
[2]Aavatsmark, I., Barkve, T., Bøe, Ø., and Mannseth, T.. Discretization on unstructured grids for inhomogeneous, anisotropic media. II. Discussion and numerical results. SIAM J. Sci. Comput, 19:1717–1736 (electronic), 1998.CrossRefGoogle Scholar
[3]Brandts, J., Korotov, S., and Kˇrížek, M.. Dissection of the path-simplex in Жn into n path-subsimplices. Lin. Alg. Appl., 421:382–393, 2007.CrossRefGoogle Scholar
[4]Brandts, J., Korotov, S., and Kˇrížek., M.The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem. Lin. Alg. Appl., 429:2344–2357, 2008.CrossRefGoogle Scholar
[5]Burman, E. and Ern, A.. Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes. C. R. Acad. Sci. Paris, Ser.I 338:641–646, 2004.CrossRefGoogle Scholar
[6]Chan, T. F. and Shen, J.. Non-texture inpainting by curvature driven diffusions (CDD). J. Vis. Commun. Image Rep, 12:436–449, 2000.Google Scholar
[7]Chan, T. F., Shen, J., and Vese, L.. Variational PDE models in image processing. Not AMS J., 50:14–26, 2003.Google Scholar
[8]Ciarlet, P. G.. Discrete maximum principle for finite difference operators. Aequationes Math., 4:338–352, 1970.CrossRefGoogle Scholar
[9]Ciarlet, P. G. and Raviart, P.-A.. Maximum principle and uniform convergence for the finite element method. Comput. Meth. Appl. Mech. Eng., 2:17–31, 1973.CrossRefGoogle Scholar
[10]Crumpton, P. I., Shaw, G. J., and Ware, A. F.. Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients. J. Comput. Phys., 116:343–358, 1995.CrossRefGoogle Scholar
[11]Drăgănescu, A., Dupont, T. F., and Scott, L. R.. Failure of the discrete maximum principle for an elliptic finite element problem. Math. Comp., 74:1–23, 2004.CrossRefGoogle Scholar
[12]Ern, A. and Guermond, J. L.. Theory and Practice of Finite Elements. Sprigner-Verlag, New York, 2004.CrossRefGoogle Scholar
[13]Ertekin, T., Abou-Kassem, J. H., and King, G. R.. Basic Applied Reservoir Simulation. SPE textbook series, Vol. 7, Richardson, Texas, 2001.CrossRefGoogle Scholar
[14]Forsyth, P. A.. A control-volume, finite-element method for local mesh refinement in thermal reservoir simulation. SPE Reservoir Engineering, 5:561–566 (Paper SPE 18415), 1990.CrossRefGoogle Scholar
[15]Gűnter, S. and Lackner, K.. A mixed implicit-explicit finite difference scheme for heat transport in magnetised plasmas. J. Comput. Phys., 228:282–293, 2009.CrossRefGoogle Scholar
[16]Gűnter, S., Lackner, K., and Tichmann, C.. Finite element and higher order difference formulations for modelling heat transport in magnetised plasmas. J. Comput. Phys., 226:2306–2316, 2007.CrossRefGoogle Scholar
[17]Gűnter, S., Yu, Q., Kruger, J., and Lackner, K.. Modelling of heat transport in magnetised plasmas using non-aligned coordinates. J. Comput. Phys., 209:354–370, 2005.CrossRefGoogle Scholar
[18]Karátson, J. and Korotov, S.. Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math., 99:669–698, 2005.CrossRefGoogle Scholar
[19]Karátson, J. and Korotov, S.. Discrete maximum principles for finite element solutions of some mixed nonlinear elliptic problems using quadratures. J. Comput. Appl. Math., 192:75–88, 2006.CrossRefGoogle Scholar
[20]Karátson, J., Korotov, S., and Kˇrížek, M.. On discrete maximum principles for nonlinear elliptic problems. Math. Comput. Sim., 76:99–108, 2007.CrossRefGoogle Scholar
[21]Karras, D. A. and Mertzios, G. B.. New PDE-based methods for image enhancement using SOM and Bayesian inference in various discretization schemes. Meas. Sci. Technol., 20:104012, 2009.CrossRefGoogle Scholar
[22]Kuzmin, D., Shashkov, M. J., and Svyatskiy, D.. A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems. J. Comput. Phys., 228:3448–3463, 2009.CrossRefGoogle Scholar
[23]Křížek, M. and Lin, Q.. On diagonal dominance of stiffness matrices in 3D. East-West J. Numer. Math., 3:59–69, 1995.Google Scholar
[24]Le Potier, C.. Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés. C. R. Math. Acad. Sci. Paris, 341:787–792, 2005.CrossRefGoogle Scholar
[25]Le Potier, C.. A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. Int. J. Finite Vol., 6:20, 2009.Google Scholar
[26]Le Potier, C.. Un schéma linéaire vérifiant le principe du maximum pour des opérateurs de diffusion très anisotropes sur des maillages déformés. C. R. Math. Acad. Sci. Paris, 347:105–110, 2009.CrossRefGoogle Scholar
[27]Letniowski, F. W.. Three-dimensional delaunay triangulations for finite element approximations to a second-order diffusion operator. SIAM J. Sci. Stat. Comput., 13:765–770, 1992.CrossRefGoogle Scholar
[28]Li, X. P. and Huang, W.. An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems. J. Comput. Phys. 229 (2010), 8072–8094. (arXiv:1003.4530v2)CrossRefGoogle Scholar
[29]Li, X. P., Svyatskiy, D., and Shashkov, M.. Mesh adaptation and discrete maximum principle for 2D anisotropic diffusion problems. Technical Report LA-UR 10-01227, Los Alamos National Laboratory, Los Alamos, NM, 2007.Google Scholar
[30]Lipnikov, K., Shashkov, M., Svyatskiy, D., and Vassilevski, Yu.. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys., 227:492–512, 2007.CrossRefGoogle Scholar
[31]Liska, R. and Shashkov, M.. Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems. Comm. Comput. Phys., 3:852–877, 2008.Google Scholar
[32]Mlacnik, M. J. and Durlofsky, L. J.. Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. J. Comput. Phys., 216:337–361, 2006.CrossRefGoogle Scholar
[33]Mumford, D. and Shah, J.. Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math, 42:577–685, 1989.CrossRefGoogle Scholar
[34]Nishikawa, K. and Wakatani, M.. Plasma Physics. Springer-Verlag Berlin Heidelberg, New York, 2000.CrossRefGoogle Scholar
[35]Perona, P. and Malik, J.. Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intel., 12:629–639, 1990.CrossRefGoogle Scholar
[36]Sharma, P. and Hammett, G.W.. Preserving monotonicity in anisotropic diffusion. J. Comput. Phys., 227:123–142, 2007.CrossRefGoogle Scholar
[37]Sommerville, D. M. Y.. An Introduction to the Geometry of n Dimensions. Methuen & Co. LTD., London, 1929.Google Scholar
[38]Stix, T.H.. Waves in Plasmas. Amer. Inst. Phys., New York, 1992.Google Scholar
[39]Stoyan, G.. On a maximum principle for matrices, and on conservation of monotonicity. With applications to discretization methods. Z. Angew. Math. Mech., 62:375–381, 1982.CrossRefGoogle Scholar
[40]Stoyan, G.. On maximum principles for monotone matrices. Lin. Alg. Appl., 78:147–161, 1986.CrossRefGoogle Scholar
[41]Strang, G. and Fix, G. J.. An Analysis of the Finite Element Method. Prentice Hall, Englewood Cliffs, NJ, 1973.Google Scholar
[42]Varga, R. S.. On a discrete maximum principle. SIAM J. Numer. Anal., 3:355–359, 1966.CrossRefGoogle Scholar
[43]Weickert, J.. Anisotropic Diffusion in Image Processing. Teubner-Verlag, Stuttgart, Germany, 1998.Google Scholar
[44]Xu, J. and Zikatanov, L.. A monotone finite element scheme for convection-diffusion equations. Math. Comput., 69:1429–1446, 1999.CrossRefGoogle Scholar