Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TWO-SCALE PRODUCT APPROXIMATION FOR SEMILINEAR PARABOLIC PROBLEMS IN MIXED METHODS

  • Kim, Dongho (University College Yonsei University) ;
  • Park, Eun-Jae (Department of Mathematics and Department of Computational Science and Engineering Yonsei University) ;
  • Seo, Boyoon (Department of Mathematics Yonsei University)
  • Received : 2013.03.25
  • Accepted : 2013.08.06
  • Published : 2014.03.01
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Abstract

We propose and analyze two-scale product approximation for semilinear heat equations in the mixed finite element method. In order to efficiently resolve nonlinear algebraic equations resulting from the mixed method for semilinear parabolic problems, we treat the nonlinear terms using some interpolation operator and exploit a two-scale grid algorithm. With this scheme, the nonlinear problem is reduced to a linear problem on a fine scale mesh without losing overall accuracy of the final system. We derive optimal order $L^{\infty}((0, T];L^2({\Omega}))$-error estimates for the relevant variables. Numerical results are presented to support the theory developed in this paper.

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References

  1. C. Bi and V. Ginting, Two-grid finite volume element method for linear and nonlinear elliptic problems, Numer. Math. 108 (2007), no. 2, 177-198. https://doi.org/10.1007/s00211-007-0115-9
  2. S. C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008.
  3. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
  4. Z. Chen, $L^p$-posteriori error analysis of mixed methods for linear and quasilinear elliptic problems, Modeling, mesh generation, and adaptive numerical methods for partial differential equations (Minneapolis, MN, 1993), 187-199, IMA Vol. Math. Appl., 75, Springer, New York, 1995.
  5. Z. Chen and J. Douglas, Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems, Mat. Apl. Comput. 10 (1991), no. 2, 137-160.
  6. Y. Chen, Y. Huang, and D. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations, Internat. J. Numer. Methods Engrg. 57 (2003), no. 2, 193-209. https://doi.org/10.1002/nme.668
  7. C. Chen, S. Larsson, and N. Zhang, Error estimates of optimal order for finite element methods with interpolated coefficients for the nonlinear heat equation, IMA J. Numer. Anal. 9 (1989), no. 4, 507-524. https://doi.org/10.1093/imanum/9.4.507
  8. I. Christie, D. F. Griffiths, A. R. Mitchell, and J. M. Sanz-Serna, Product approximation for nonlinear problems in the finite element method, IMA J. Numer. Anal. 1 (1981), no. 3, 253-266. https://doi.org/10.1093/imanum/1.3.253
  9. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, 1978.
  10. C. Dawson and M. F. Wheeler, Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993), 191-203, Contemp. Math., 180, Amer. Math. Soc., Providence, RI, 1994.
  11. J. Douglas, Jr. and J. E. Roberts, Global Estimates for Mixed Methods for second Order Elliptic Equations, Math. Comp. 44 (1985), no. 169, 39-52. https://doi.org/10.1090/S0025-5718-1985-0771029-9
  12. J. Douglas, Jr. and T. Dupont, The effect of interpolating the coefficients in nonlinear parabolic Galerkin procedures, Math. Comp. 20 (1975), no. 130, 360-389.
  13. J. Douglas, Jr., T. Dupont, and R. E. Ewing, Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. Numer. Anal. 16 (1979), no. 3, 503-522. https://doi.org/10.1137/0716039
  14. J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numer. 17 (1983), no. 1, 17-33. https://doi.org/10.1051/m2an/1983170100171
  15. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, 2004.
  16. J. Jin, S. Shu, and J. Xu, A two-grid discretization method for decoupling systems of partial differential equations, Math. Comp. 75 (2006), no. 256, 1617-1626. https://doi.org/10.1090/S0025-5718-06-01869-2
  17. S. Larsson, V. Thomee, and N. Zhang, Interpolation of coefficients and transformation of the dependent variable in finite element methods for the nonlinear heat equation, Math. Meth. Appl. Sci. 11 (1989), no. 1, 105-124. https://doi.org/10.1002/mma.1670110108
  18. W. Layton and W. Lenferink, Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comput. 69 (1995), no. 2-3, 263-274. https://doi.org/10.1016/0096-3003(94)00134-P
  19. D. Kim and E.-J. Park, A posteriori error estimator for expanded mixed hybrid methods, Numer. Methods Partial Differential Equations 23 (2007), no. 2, 330-349. https://doi.org/10.1002/num.20178
  20. D. Kim and E.-J. Park, A Priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations, SIAM J. Numer. Anal. 48 (2010), no. 3, 1186-1207. https://doi.org/10.1137/090747002
  21. M.-Y. Kim, F. A. Milner, and E.-J. Park, Some observations on mixed methods for fully nonlinear parabolic problems in divergence form, Appl. Math. Lett. 9 (1996), no. 1, 75-81.
  22. M.-Y. Kim, E.-J. Park, S. G. Thomas, and M. F. Wheeler, A multiscale mortar mixed finite element method for slightly compressible flows in porous media, J. Korean Math. Soc. 44 (2007), no. 5, 1103-1119. https://doi.org/10.4134/JKMS.2007.44.5.1103
  23. F. A. Milner and E.-J. Park, A mixed finite element method for a strongly nonlinear second-order elliptic problem, Math. Comp. 64 (1995), no. 211, 973-988. https://doi.org/10.1090/S0025-5718-1995-1303087-3
  24. E.-J. Park, Mixed finite element methods for nonlinear second order elliptic problems, SIAM J. Numer. Anal. 32 (1995), no. 3, 865-885. https://doi.org/10.1137/0732040
  25. P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), 292-315. Lecture Notes in Math. 606, Springer, Berlin, 1977.
  26. J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, Handbook of numerical analysis, Vol. II, 523-639, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991.
  27. T. Russell and M. F. Wheeler, Finite element and finite difference methods for continuous flows in porous media, in The Mathematics of Reservoir Simulation, 35-106, R. E. Ewing, ed., Frontiers in Applied Mathematics 1, Society for Industrial and Applied Mathematics, Philadelphia, 1984.
  28. J. M. Sanz-Serna and L. Abia, Interpolation of the coefficients in nonlinear elliptic Galerkin procedures, SIAM J. Numer. Anal. 21 (1984), no. 1, 77-83. https://doi.org/10.1137/0721004
  29. A. Weiser and M. F. Wheeler, On convergence of block-centered finite differences for elliptic problems, SIAM J. Numer. Anal. 25 (1988), no. 2, 351-375. https://doi.org/10.1137/0725025
  30. L. Wu and M. B. Allen, A two-grid method for mixed finite-element solution of reaction-diffusion equations, Numer. Methods Partial Differential Equations 15 (1999), no. 3, 317-332. https://doi.org/10.1002/(SICI)1098-2426(199905)15:3<317::AID-NUM4>3.0.CO;2-U
  31. J. Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231-237. https://doi.org/10.1137/0915016
  32. J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759-1777. https://doi.org/10.1137/S0036142992232949

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