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Finite Difference Approximation of Parabolic Problems

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Part of the Springer Series in Computational Mathematics book series (SSCM,volume 46)

Abstract

Chapter 3 is concerned with the construction and the convergence analysis of finite difference schemes for parabolic initial-boundary-value problems. The central contribution of the chapter is the derivation of optimal-order bounds on the error between the analytical solution and its finite difference approximation for parabolic equations with variable coefficients under minimal regularity hypotheses on the coefficients and the solution, the minimal regularity hypotheses on the coefficients being expressed in terms of spaces of multipliers in anisotropic Sobolev spaces.

Keywords

  • Parabolic Equation
  • Error Bound
  • Time Level
  • Finite Difference Scheme
  • Parabolic Problem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 3.1
Fig. 3.2

References

  1. Amosov, A.A.: Global solvability of a nonlinear nonstationary problem with a nonlocal boundary condition of radiation heat transfer type. Differ. Equ. 41(1), 96–109 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Amosov, A.A., Zlotnik, A.A.: Difference schemes of second-order of accuracy for the equations of the one-dimensional motion of a viscous gas. USSR Comput. Math. Math. Phys. 27(4), 46–57 (1987)

    CrossRef  MATH  Google Scholar 

  3. Barbeiro, S., Ferreira, J.A., Pinto, L.: H 1-second order convergent estimates for non Fickian models. Appl. Numer. Math. 61, 201–215 (2011)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Bojović, D., Jovanović, B.S.: Application of interpolation theory to determination of convergence rate for finite difference schemes of parabolic type. Mat. Vestn. 49, 99–107 (1997)

    MATH  Google Scholar 

  5. Bojović, D., Jovanović, B.S.: Convergence of finite difference method for the parabolic problem with concentrated capacity and variable operator. J. Comput. Appl. Math. 189(1–2), 286–303 (2006)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Brenner, P., Thomée, V., Wahlbin, L.B.: Besov Spaces and Applications to Difference Methods for Initial Value Problems. Lecture Notes in Mathematics, vol. 434. Springer, Berlin (1975)

    MATH  Google Scholar 

  7. Carter, R., Giles, M.B.: Sharp error estimates for discretizations of the 1D convection-diffusion equation with Dirac initial data. IMA J. Numer. Anal. 27, 406–425 (2007)

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Datta, A.K.: Biological and Bioenvironmental Heat and Mass Transfer. Dekker, New York (2002)

    CrossRef  Google Scholar 

  9. Douglas, J., Dupont, T.: A finite element collocation method for quasilinear parabolic equations. Math. Comput. 27, 17–28 (1973)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Douglas, J., Dupont, T., Wheeler, M.F.: A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations. Math. Comput. 32, 345–362 (1978)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Dražić, M.: Convergence rates of difference approximations to weak solutions of the heat transfer equation. Technical Report 86/22, Oxford University Computing Laboratory, Numerical Analysis Group, Oxford (1986)

    Google Scholar 

  12. Druet, P.E.: Weak solutions to a time-dependent heat equation with nonlocal radiation condition and right hand side in L p (p≥1). WIAS Preprint 1253 (2008)

    Google Scholar 

  13. Givoli, D.: Exact representation on artificial interfaces and applications in mechanics. Appl. Mech. Rev. 52, 333–349 (1999)

    CrossRef  Google Scholar 

  14. Givoli, D.: Finite element modeling of thin layers. Comput. Model. Eng. Sci. 5(6), 497–514 (2004)

    Google Scholar 

  15. Godev, K.N., Lazarov, R.D.: Error estimates of finite-difference schemes in L p -metrics for parabolic boundary value problems. C. R. Acad. Bulgare Sci. 37, 565–568 (1984)

    MathSciNet  MATH  Google Scholar 

  16. Hackbusch, W.: Optimal H p,p/2 error estimates for a parabolic Galerkin method. SIAM J. Numer. Anal. 18, 681–692 (1981)

    MathSciNet  CrossRef  MATH  Google Scholar 

  17. Haroske, D.D., Triebel, H.: Distributions, Sobolev Spaces, Elliptic Equations. European Mathematical Society, Zürich (2008)

    MATH  Google Scholar 

  18. Ivanović, L.D., Jovanović, B.S., Süli, E.: On the rate of convergence of difference schemes for the heat transfer equation on the solutions from \(W_{2}^{s, s/2}\). Mat. Vesn. 36, 206–212 (1984)

    MATH  Google Scholar 

  19. Jovanović, B.S.: Convergence of projection-difference schemes for the heat equation. Mat. Vesn. 6(19) (34), 279–292 (1982). (Russian)

    Google Scholar 

  20. Jovanović, B.S.: On the convergence of finite-difference schemes for parabolic equations with variable coefficients. Numer. Math. 54, 395–404 (1989)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Jovanović, B.S.: Convergence of finite-difference schemes for parabolic equations with variable coefficients. Z. Angew. Math. Mech. 71, 647–650 (1991)

    MathSciNet  Google Scholar 

  22. Jovanović, B.S., Vulkov, L.G.: Operator approach to the problems with concentrated factors. In: Numerical Analysis and Its Applications, Rousse, 2000. Lecture Notes in Comput. Sci., pp. 439–450. Springer, Berlin (2001)

    CrossRef  Google Scholar 

  23. Jovanović, B.S., Vulkov, L.G.: On the convergence of finite difference schemes for the heat equation with concentrated capacity. Numer. Math. 89(4), 715–734 (2001)

    MathSciNet  CrossRef  MATH  Google Scholar 

  24. Jovanović, B.S., Vulkov, L.G.: Stability of difference schemes for parabolic equations with dynamical boundary conditions and conditions on conjugation. Appl. Math. Comput. 163, 849–868 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  25. Jovanović, B.S., Vulkov, L.G.: Energy stability for a class of two-dimensional interface linear parabolic problems. J. Math. Anal. Appl. 311, 120–138 (2005)

    MathSciNet  CrossRef  MATH  Google Scholar 

  26. Jovanović, B.S., Vulkov, L.G.: On the convergence of difference schemes for parabolic problems with concentrated data. Int. J. Numer. Anal. Model. 5(3), 386–406 (2008)

    MathSciNet  MATH  Google Scholar 

  27. Jovanović, B.S., Vulkov, L.G.: Numerical solution of a two-dimensional parabolic transmission problem. Int. J. Numer. Anal. Model. 7(1), 156–172 (2010)

    MathSciNet  Google Scholar 

  28. Jovanović, B.S., Vulkov, L.G.: Numerical solution of a parabolic transmission problem. IMA J. Numer. Anal. 31, 233–253 (2011)

    MathSciNet  CrossRef  MATH  Google Scholar 

  29. Jovanović, B.S., Ivanović, L.D., Süli, E.: On the convergence rate of difference schemes for the heat transfer equation. In: Vrdoljak, B. (ed.) IV Conference on Applied Mathematics, Proc. Conf. Held in Split 1984, University of Split, Split pp. 41–44 (1985)

    Google Scholar 

  30. Junkosa, M.L., Young, D.M.: On the order of convergence of solutions of a difference equation to a solution of the diffusion equation. SIAM J. 1, 111–135 (1953)

    Google Scholar 

  31. Kačur, J., Van Keer, R., West, J.: On the numerical solution to a semi-linear transient heat transfer problem in composite media with nonlocal transmission conditions. In: Lewis, R.W. (ed.) Numerical Methods in Thermal Problems, VIII, pp. 1508–1519. Pineridge Press, Swansea (1993)

    Google Scholar 

  32. Kuzik, A.M., Makarov, V.L.: The rate of convergence of a difference scheme using the sum approximation method for generalized solutions. USSR Comput. Math. Math. Phys. 26(3), 192–196 (1986)

    CrossRef  Google Scholar 

  33. Lazarov, R.D.: Convergence of difference method for parabolic equations with generalized solutions. Pliska Stud. Math. Bulg. 5, 51–59 (1982). (Russian)

    MathSciNet  Google Scholar 

  34. Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications. Dunod, Paris (1968)

    MATH  Google Scholar 

  35. Melenk, J.M.: On approximation in meshless methods. In: Craig, A.W., Blowey, J.F. (eds.) Frontiers of Numerical Analysis, Durham, 2004. Universitext. Springer, Berlin (2005)

    Google Scholar 

  36. Qatanani, N., Barham, A., Heeh, Q.: Existence and uniqueness of the solution of the coupled conduction-radiation energy transfer on diffusive-gray surfaces. Surv. Math. Appl. 2, 43–58 (2007)

    MathSciNet  MATH  Google Scholar 

  37. Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numer. Math. 43, 309–327 (1984)

    MathSciNet  CrossRef  MATH  Google Scholar 

  38. Samarskiĭ, A.A.: The Theory of Difference Schemes. Nauka, Moscow (1983). (Russian); English edn.: Monographs and Textbooks in Pure and Applied Mathematics, vol. 240. Dekker, New York (2001)

    Google Scholar 

  39. Samarskiĭ, A.A., Iovanovich, B.S., Matus, P.P., Shcheglik, V.S.: Finite-difference schemes on adaptive time grids for parabolic equations with generalized solutions. Differ. Equ. 33, 981–990 (1997)

    Google Scholar 

  40. Scott, J.A., Seward, W.L.: Finite difference methods for parabolic problems with nonsmooth initial data. Technical Report 86/22, Oxford University Computing Laboratory, Numerical Analysis Group, Oxford (1987)

    Google Scholar 

  41. Seward, W.L., Kasibhatla, P.S., Fairweather, G.: On the numerical solution of a model air pollution problem with non-smooth initial data. Commun. Appl. Numer. Methods 6, 145–156 (1990)

    CrossRef  MATH  Google Scholar 

  42. Süli, E., Mayers, D.F.: An Introduction to Numerical Analysis, 2nd edn. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  43. Thomée, V.: Negative norm estimates and superconvergence in Galerkin methods for parabolic problems. Math. Comput. 34, 93–113 (1980)

    CrossRef  MATH  Google Scholar 

  44. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin (2006)

    MATH  Google Scholar 

  45. Thomée, V., Wahlbin, L.B.: Convergence rates of parabolic difference schemes for non-smooth data. Math. Comput. 28(125), 1–13 (1974)

    CrossRef  MATH  Google Scholar 

  46. Tikhonov, A.N.: On functional equations of the Volterra type and their applications to some problems of mathematical physics. Byull. Mosc. Gos. Univ., Ser. Mat. Mekh. 1(8), 1–25 (1938). (Russian)

    MathSciNet  Google Scholar 

  47. Weinelt, W., Lazarov, R.D., Streit, U.: Order of convergence of finite-difference schemes for weak solutions of the heat conduction equation in an anisotropic inhomogeneous medium. Differ. Equ. 20, 828–834 (1984)

    MathSciNet  MATH  Google Scholar 

  48. Wheeler, M.F.: L estimates of optimal orders for Galerkin methods for one dimensional second order parabolic and hyperbolic problems. SIAM J. Numer. Anal. 10, 908–913 (1973)

    MathSciNet  CrossRef  MATH  Google Scholar 

  49. Wloka, J.: Partial Differential Equations. Cambridge Univ. Press, Cambridge (1987)

    CrossRef  MATH  Google Scholar 

  50. Zlamal, M.: Finite element methods for parabolic equations. Math. Comput. 28, 393–404 (1974)

    MathSciNet  CrossRef  MATH  Google Scholar 

  51. Zlotnik, A.A.: Convergence rate estimate in L 2 of projection-difference schemes for parabolic equations. USSR Comput. Math. Math. Phys. 18(6), 92–104 (1978)

    MathSciNet  CrossRef  Google Scholar 

  52. Zlotnik, A.A.: Estimation of the rate of convergence in V 2(Q T ) of projection-difference schemes for parabolic equations. Mosc. Univ. Comput. Math. Cybern. 1, 28–38 (1980). 1980

    Google Scholar 

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Jovanović, B.S., Süli, E. (2014). Finite Difference Approximation of Parabolic Problems. In: Analysis of Finite Difference Schemes. Springer Series in Computational Mathematics, vol 46. Springer, London. https://doi.org/10.1007/978-1-4471-5460-0_3

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