Abstract
A general setting for a new approach to constructing vector splitting schemes in mixed FEM for heat transfer problem is considered. For space approximation Raviart–Thomas finite elements of lowest order are implemented on rectangular (parallelepiped) mesh. The main question discussed is how to implement time discretization so as to obtain efficient numerical algorithms.
The key idea of the proposed approach is to use scalar splitting schemes for heat flux divergence. This allows one to carry out accuracy and stability analysis on the basis of the well-known results for underlying scalar splitting schemes. A priori estimates are obtained for vector splitting schemes in two- and three-dimensional cases using the idea of flux decomposition onto divergence-free and potential components at discrete level. It is shown that splitting schemes which have been proposed by other approaches based on Uzawa algorithm’s modification and block triangular factorization are included as special cases by an appropriate choice of the underlying splitting schemes for heat flux divergence. Generalization of the discussed approach is presented for second-order hyperbolic problems. Moreover, the general idea of using splitting schemes for the flux divergence to construct splitting schemes for fluxes can be easily extended to first-order hyperbolic systems, but by now several questions still remain unsolved.
Funding source: Russian Science Foundation
Award Identifier / Grant number: 15-11-10024
Funding statement: This work is supported by Russian Science Foundation grant No. 15-11-10024.
References
[1] T. Arbogast, C.-S. Huang, and S.-M. Yang, Improved accuracy for alternating-direction methods for parabolic equations based on regular and mixed finite elements, Math. Models Meth. Appl. Sci. 17(8) (2007), 1279–1305.10.1142/S0218202507002285Search in Google Scholar
[2] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.10.1007/978-1-4612-3172-1Search in Google Scholar
[3] J. Douglas, Jr. and J. E. Gunn, A general formulation of alternating direction methods, Part 1. Hyperbolic and parabolic problems, Numer. Math. 6 (1964), 428–453.10.1007/BF01386093Search in Google Scholar
[4] J. Douglas, Jr. and P. Pietra, A description of some alternating-direction iterative techniques for mixed finite element methods, Math. Comp. Meth. Seismic Explor. Reservoir Model. (Ed. W. E. Fitzgibbon), SIAM, Philadelphia, PA (1986), 37–53.Search in Google Scholar
[5] J. Douglas, Jr. and J. Wang, Superconvergence for mixed finite element methods on rectangular domains, Calcolo26 (1989), 121–134.10.1007/BF02575724Search in Google Scholar
[6] R. Duran, Superconvergence for rectangular mixed finite elements, Numer. Math., 58 (1990), 287–298.10.1007/BF01385626Search in Google Scholar
[7] G. I. Habetler and E.L. Wachspress, Symmetric successive overrelaxation in solving diffusion difference equations, Math. Comp. 15 (1961), 356–362.10.1090/S0025-5718-1961-0129139-9Search in Google Scholar
[8] G. I. Marchuk, Methods of Computational Mathematics, Nauka, Moscow, 1980.Search in Google Scholar
[9] G. I. Marchuk, Splitting and alternating direction methods, Handbook of Numerical Analysis (Eds. P.G. Ciarlet and J. L. Lions), 1 (1990), 197–462.10.1016/S1570-8659(05)80035-3Search in Google Scholar
[10] G. I. Marchuk, Methods of Splitting, Nauka, Moscow, 1988 (in Russian).Search in Google Scholar
[11] M.Nakata, A. Weiser, and M. F.Wheeler, Some superconvergence results for mixed finite element methods for elliptic problems on rectangular domains, The Mathematics of Finite Elements and Applications, Vol. V (Ed. J. R. Whiteman), Academic Press, London, 1985.10.1016/B978-0-12-747255-3.50032-0Search in Google Scholar
[12] D.W. Peaceman and H. Rachford, The numerical solution of parabolic and elliptic equations, J. Soc. Indust. Appl. Math. 3 (1955), 28–41.10.1137/0103003Search in Google Scholar
[13] R.A. Raviart and J. M.Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. (Eds. I. Galligani and E. Magenes), Springer-Verlag, New York, 1977, pp. 292– 315.10.1007/BFb0064470Search in Google Scholar
[14] R.D. Richtmyer and K.W. Morton, Difference Methods for Initial-Value Problems, Wiley, 1967.Search in Google Scholar
[15] A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York, 2001.10.1201/9780203908518Search in Google Scholar
[16] P. N. Vabishchevich, Additive Operator-Difference Schemes: Splitting Schemes, de Gruyter, Berlin–Boston, 2014.10.1515/9783110321463Search in Google Scholar
[17] P. N. Vabishchevich, Flux splitting schemes for parabolic problems, Comp. Math. Math. Phys. 52 (2012), 1128–1138.10.1007/978-3-642-41515-9_12Search in Google Scholar
[18] K. V. Voronin and Yu. M. Laevsky, On splitting schemes in the mixed finite element method, Numer. Anal. Appl. 5(2) (2012), 150–155.10.1134/S1995423912020085Search in Google Scholar
[19] K. V. Voronin and Yu. M. Laevsky, Splitting schemes in the mixed finite-element method for the solution of heat transfer problems Math. Models Comp. Simul. 5(2) (2013), 167–174.10.1134/S2070048213020105Search in Google Scholar
[20] K. V. Voronin and Yu. M. Laevsky, On the stability of some flux splitting schemes, Numer. Anal. Appl., 8(2) 2015, 113–121.10.1134/S1995423915020032Search in Google Scholar
[21] N. N. Yanenko, The Method of Fractional Steps, Springer-Verlag, New York, 1971.10.1007/978-3-642-65108-3Search in Google Scholar
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