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QUADRATURE BASED FINITE ELEMENT METHODS FOR LINEAR PARABOLIC INTERFACE PROBLEMS

  • Deka, Bhupen (Department of Mathematics Indian Institute of Technology Guwahati) ;
  • Deka, Ram Charan (Department of Mathematical Sciences Tezpur University)
  • Received : 2013.03.09
  • Published : 2014.05.31

Abstract

We study the effect of numerical quadrature in space on semidiscrete and fully discrete piecewise linear finite element methods for parabolic interface problems. Optimal $L^2(L^2)$ and $L^2(H^1)$ error estimates are shown to hold for semidiscrete problem under suitable regularity of the true solution in whole domain. Further, fully discrete scheme based on backward Euler method has also analyzed and optimal $L^2(L^2)$ norm error estimate is established. The error estimates are obtained for fitted finite element discretization based on straight interface triangles.

Keywords

References

  1. I. Babuska, The finite element method for elliptic equations with discontinuous coefficients, Computing 5 (1970), 207-213. https://doi.org/10.1007/BF02248021
  2. J. W. Barrett and C. M. Elliott, Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA J. Numer. Anal. 7 (1987), no. 3, 283-300. https://doi.org/10.1093/imanum/7.3.283
  3. J. H. Bramble and J. T. King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math. 6 (1996), no. 2, 109-138. https://doi.org/10.1007/BF02127700
  4. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
  5. C. M. Chen and V. Thomee, The lumped mass finite element method for a parabolic problems, J. Austral. Math. Soc. Ser. B 26 (1985), no. 3, 329-354. https://doi.org/10.1017/S0334270000004549
  6. Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998), no. 2, 175-202. https://doi.org/10.1007/s002110050336
  7. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1975.
  8. B. Deka, Finite element methods with numerical quadrature for elliptic problems with smooth interfaces, J. Comput. Appl. Math. 234 (2010), no. 2, 605-612. https://doi.org/10.1016/j.cam.2009.12.052
  9. B. Deka and T. Ahmed, Semidiscrete finite element methods for linear and semilinear parabolic problems with smooth interfaces: some new optimal error estimates, Numer. Funct. Anal. Optim. 33 (2012), no. 5, 524-544. https://doi.org/10.1080/01630563.2011.651189
  10. B. Deka and R. K. Sinha, $L^{\infty}(L^2)$ and $L^{\infty}(H^1)$ norms error estimates in finite element method for linear parabolic interface problems, Numer. Funct. Anal. Optim. 32 (2011), no. 3, 267-285. https://doi.org/10.1080/01630563.2010.532272
  11. B. Deka, R. K. Sinha, R. C. Deka, and T. Ahmed, Finite element method with quadrature for parabolic interface problems, Neural Parallel Sci. Comput. 21 (2013), no. 3-4, 477-496.
  12. H. Duan, P. Lin, and Roger C. E. Tan, Analysis of a continuous finite element method for H(curl, div)-elliptic interface problem, Numer. Math. 123 (2013), no. 4, 671-707. https://doi.org/10.1007/s00211-012-0500-x
  13. A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 47-48, 5537-5552. https://doi.org/10.1016/S0045-7825(02)00524-8
  14. J. Huang and J. Zou, A mortar element method for elliptic problems with discontinuous coefficients, IMA J. Numer. Anal. 22 (2002), no. 4, 554-576.
  15. J. Huang and J. Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, J. Differential Equations 184 (2002), no. 2, 570-586. https://doi.org/10.1006/jdeq.2001.4154
  16. M. Kumar and P. Joshi, Some numerical techniques for solving elliptic interface problems, Numer. Methods Partial Differential Equations 28 (2012), no. 1, 94-114. https://doi.org/10.1002/num.20609
  17. O. A. Ladyzhenskaya, V. Ja. Rivkind, and N. N. Ural'ceva, The classical solvability of diffraction problems, Trudy Mat. Inst. Steklov 92 (1966), 116-146.
  18. J. Li, J. M. Melenk, B. Wohlmuth, and J. Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems, Appl. Numer. Math. 60 (2010), no. 1-2, 19-37. https://doi.org/10.1016/j.apnum.2009.08.005
  19. R. Massjung, An unfitted discontinuous Galerkin method applied to elliptic interface problems, SIAM J. Numer. Anal. 50 (2012), no. 6, 3134-3162. https://doi.org/10.1137/090763093
  20. B. F. Nielsen, Finite element discretizations of elliptic problems in the presence of arbitrarily small ellipticity: An error analysis, SIAM J. Numer. Anal. 36 (1999), no. 2, 368-392. https://doi.org/10.1137/S0036142997319431
  21. P. A. Raviart, The Use of Numerical Integration in Finite Element Methods for Solving Parabolic Equations, Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972), pp. 233-264. Academic Press, London, 1973.
  22. R. K. Sinha and B. Deka, Optimal error estimates for linear parabolic problems with discontinuous coefficients, SIAM J. Numer. Anal. 43 (2005), no. 2, 733-749. https://doi.org/10.1137/040605357
  23. R. K. Sinha and B. Deka, A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems, Calcolo 43 (2006), no. 4, 253-278. https://doi.org/10.1007/s10092-006-0122-8
  24. V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, 1997.

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