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A Constrained Finite Element Method Based on Domain Decomposition Satisfying the Discrete Maximum Principle for Diffusion Problems

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  30 July 2015

Xingding Chen  and
Guangwei Yuan
Xingding Chen*
Affiliation:
Department of Mathematics, School of Science, Beijing Technology and Business University, Beijing 100048, P.R. China
Guangwei Yuan
Affiliation:
LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China
*
*Corresponding author. Email addresses: [email protected] (X. D. Chen), [email protected] (G. W. Yuan)
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Abstract

In this paper, we are concerned with the constrained finite element method based on domain decomposition satisfying the discrete maximum principle for diffusion problems with discontinuous coefficients on distorted meshes. The basic idea of domain decomposition methods is used to deal with the discontinuous coefficients. To get the information on the interface, we generalize the traditional Neumann-Neumann method to the discontinuous diffusion tensors case. Then, the constrained finite element method is used in each subdomain. Comparing with the method of using the constrained finite element method on the global domain, the numerical experiments show that not only the convergence order is improved, but also the nonlinear iteration time is reduced remarkably in our method.

Keywords

Discrete maximum principlediscontinuous coefficientsdomain decompositionconstrained finite element method

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 2 , August 2015 , pp. 297 - 320
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Burman, E. and Ern, A., Discrete maximum priciple for Galerkin approximation of the Laplace operator on arbitrary meshes, Comptes Rendus Mathematique Academie des Sciences. Paris, 338 (2004), 641–646.Google Scholar
[2]Ciarlet, P. G., Discrete maximum principle for finite-difference operators, Aeq. Math. 4 (1970), 338-352.Google Scholar
[3]Ciarlet, P. G. and Raviart, P. A., Maximum principle and convergence for the finite element method, Comput. Methods Appl. Mech. Eng. 2 (1973), 17-31.Google Scholar
[4]Varga, R. S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, 1962.Google Scholar
[5]Liska, R. and Shashkov, M., Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems, Comm. Comput. Phys., 3 (2008), 852-877.Google Scholar
[6]Nakshatrala, K. and Valocchi, A., Non-negative mixed finite element formulation for a tensorial diffusion equation, J. Comput. Phy., 228 (2009), 6726-6752.Google Scholar
[7]Nagarajan, H. and Nakshatrala, K., Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids, Int. J. Numer. Meth. Flu., 67 (2011), 820-847.Google Scholar
[8]Kuzmin, D., Shashkov, M. and Svyatskiy, D., A constrained finite element method satisfying the discrete maximum priciple for anisotropic diffusion problems, J. Comput. Phys., 228 (2009), 3448-3463.CrossRefGoogle Scholar
[9]Li, D. Y., Shui, H. S. and Tang, M. J., On the finite difference scheme of two-dimensional parabolic equation in a non-recangular mesh, Int. J. Numer. Meth. Comput. Appl., 1 (1980), 217-224.Google Scholar
[10]Huang, W. Z. and Kappen, A., A study of cell-center finite volume methods for diffusion equations, Mathematics Research Report, University of Kansas, Lawrence KS66045, 98-10-01.Google Scholar
[11]Aavatsmark, I., An introduction to multipoint flux approximations for quadrilateral grids, Computational Geosciences; 6 (2002), 405–432.Google Scholar
[12]Friis, H. and Ewards, M., A family of mp finite-volume schemes with full presseure support for general tensor oressure equation on cell-centered triangular grids, J. Comput.Phys., 230 (2011), 205–231.Google Scholar
[13]Nordbotten, J. and Eigestad, G., Discritization on quadrilateral grids with improved monotoncity properties, J. Comput. Phys., 203 (2005), 744–760.Google Scholar
[14]Lipnikov, K., Shashkov, M. and Svyatskiy, D., The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes, J. Comput. Phys., 211 (2006), 473–491.Google Scholar
[15]Lipnikov, K., Shashkov, M. and Yotov, I., Local flux mimetic finite difference methods, Numer. Math., 112 (2009), 115–152.Google Scholar
[16]Protter, M. H. and Weinberger, H. F., Maximum principlrs in Differential Equations, Prentice-Hall, 1967.Google Scholar
[17]Karatson, J. and Korotov, S., Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numer. Math. 99 (2005), 669–698.Google Scholar
[18]Toselli, A. and Widlund, O., Domain Decomposition Methods-Algorithms And Theory, Springer, Berlin, 2004.Google Scholar
[19]Kuzmin, D. and Moller, M., Algebraic flux correction I. Scalar conservation laws, in: Kuzmin, D., Lohner, R., Turek, S. (Eds.), Flux-Corrected Transport: Principles Algorithms and Applications, Springer Berlin, 2005, 155–206.Google Scholar
[20]Kuzmin, D., On the design of general-purpose flux limiters for implicit FEM with a consistent mass matrix. I. Scalar convection, J. Comput. Phys. 219 (2006), 513–531.Google Scholar
[21]Sheng, Z. Q. and Yuan, G. W., A nine point scheme for the approxiamtion of diffusion operators on distorted quadrilateral meshes, SIAM J. Sci. Comput., 30 (2008), 1341-1361.Google Scholar
[22]Yuan, G. W. and Sheng, Z. Q., Analysis of accuracy of a finite volume scheme for diffusion equations on distorted meshes, J. Comput. Phy., 224 (2007), 1170-1189.Google Scholar
[23]Lv, T., Shi, J. M. and Lin, Z. B., Domain Decomposition Methods-The New Technique For The Numerical Analysis Of Partial Differential Equations, Science Press, China, 1992.Google Scholar
[24]Yuan, G. W. and Sheng, Z. Q., Monotone finite volume schemes for diffusion equations on polygonal meshes, J. Comput. Phy., 227 (2008), 6288-6312.CrossRefGoogle Scholar