About DML-CZ | FAQ | News | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Jia, Shanghui ; Li, Deli ; Liu, Tang ; Zhang, Shuhua
Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media. (English). Applications of Mathematics, vol. 53 (2008), issue 1, pp. 13-39
MSC: 45K05, 65M12, 65M60, 65R20, 76M10, 76S05 | MR 2382288 | Zbl 1177.76408
Keywords:
integro-differential equations; mixed finite element methods; mixed regularized Green’s functions; asymptotic expansions; interpolation defect correction; interpolation postprocessing; a posteriori error estimators
Summary:

References:
[1] I.  Babuška: The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973), 179–192. MR 0359352
[2] H.  Blum: Asymptotic Error Expansion and Defect in the Finite Element Method. University of Heidelberg, Institut für Angewandte Mathematik, Heidelberg, .
[3] H.  Blum, Q.  Lin, R.  Rannacher: Asymptotic error expansion and Richardson extrapolation for linear finite elements. Numer. Math. 49 (1986), 11–38. MR 0847015
[4] F.  Brezzi: On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO, Anal. Numér. 2 (1974), 129–151. MR 0365287 | Zbl 0338.90047
[5] H.  Brunner, Y.  Lin, S.  Zhang: Higher accuracy methods for second-kind Volterra integral equations based on asymptotic expansions of iterated Galerkin methods. J.  Integral Equations Appl. 10 (1998), 375–396. MR 1669667
[6] J. R.  Cannon, Y.  Lin: Non-classical $H^1$  projection and Galerkin methods for nonlinear parabolic integro-differential equations. Calcolo 25 (1988), 187–201. MR 1053754
[7] J. R.  Cannon Y.  Lin: A priori $L^2$ error estimates for finite element methods for nonlinear diffusion equations with memory. SIAM J.  Numer. Anal. 27 (1990), 595–607. MR 1041253
[8] C.  Chen, Y.  Huang: Higher Accuracy Theory of FEM. Hunan Science Press, Changsha, 1995.
[9] J.  Douglas, Jr., J. E.  Roberts: Global estimates for mixed methods for second order elliptic equations. Math. Comput. 44 (1985), 39–52. MR 0771029
[10] R. E.  Ewing, Y.  Lin, T.  Sun, J.  Wang, S. Zhang: Sharp $L^2$  error estimates and superconvergence of mixed finite element methods for nonFickian flows in porous media. SIAM J.  Numer. Anal. 40 (2002), 1538–1560. MR 1951906
[11] R. E.  Ewing, Y.  Lin, J.  Wang: A numerical approximation of nonFickian flows with mixing length growth in porous media. Acta Math. Univ. Comenian. (N.  S.) 70 (2001), 75–84. MR 1865361
[12] R. E.  Ewing, Y.  Lin, J.  Wang: A backward Euler method for mixed finite element approximations of nonFickian flows with non-smooth data in porous media. Preprint.
[13] R. E.  Ewing, Y.  Lin, J. Wang, S.  Zhang: $L^{\infty }$-error estimates and superconvergence in maximum norm of mixed finite element methods for nonFickian flows in porous media. Int. J.  Numer. Anal. Model. 2 (2005), 301–328. MR 2112650
[14] G.  Fairweather, Q.  Lin, Y.  Lin, J.  Wang, S.  Zhang: Asymptotic expansions and Richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods. SIAM J. Numer. Anal. 44 (2006), 1122–1149. MR 2231858
[15] P.  Helfrich: Asymptotic expansion for the finite element approximations of parabolic problems. Bonn. Math. Schr. 158 (1984), 11–30. MR 0793413
[16] S.  Jia, D.  Li, S.  Zhang: Asymptotic expansions and Richardson extrapolation of approximate solutions for integro-differential equations by mixed finite element methods. Adv. Comput. Math (to appear). MR 2447252
[17] M. N.  LeRoux, V.  Thomée: Numerical solutions of semilinear integro-differential equations of parabolic type with nonsmooth data. SIAM J. Numer. Anal. 26 (1989), 1291–1309.
[18] Q.  Lin, I. H.  Sloan, R.  Xie: Extrapolation of the iterated-collocation method for integral equations of the second kind. SIAM J.  Numer. Anal. 27 (1990), 1535–1541. MR 1080337
[19] Q.  Lin, N.  Yan: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Publishers, , 1996.
[20] Q.  Lin, S.  Zhang: An immediate analysis for global superconvergence for integrodifferential equations. Appl. Math. 42 (1997), 1–21. MR 1426677
[21] Q.  Lin, S.  Zhang, N.  Yan: Asymptotic error expansion and defect correction for Sobolev and viscoelasticity type equations. J.  Comput. Math. 16 (1998), 57–62. MR 1606093
[22] Q.  Lin, S.  Zhang, N.  Yan: High accuracy analysis for integrodifferential equations. Acta Math. Appl. Sin. 14 (1998), 202–211. MR 1620823
[23] Q.  Lin, S.  Zhang, N.  Yan: Methods for improving approximate accuracy for hyperbolic integro-differential equations. Syst. Sci. Math. Sci. 10 (1997), 282–288. MR 1469188
[24] Q.  Lin, S.  Zhang, N.  Yan: Extrapolation and defect correction for diffusion equations with boundary integral conditions. Acta Math. Sci. 17 (1997), 409–412. MR 1613231
[25] Q.  Lin, S.  Zhang, N.  Yan: An acceleration method for integral equations by using interpolation post-processing. Adv. Comput. Math. 9 (1998), 117–128. MR 1662762
[26] T.  Lin, Y.  Lin, M.  Rao, S.  Zhang: Petrov-Galerkin methods for linear Volterra integro-differential equations. SIAM J.  Numer. Anal. 38 (2000), 937–963. MR 1781210
[27] Y.  Lin: On maximum norm estimates for Ritz-Volterra projections and applications to some time-dependent problems. J.  Comput. Math. 15 (1997), 159–178. MR 1448820
[28] Y.  Lin, V.  Thomée, L.  Wahlbin: Ritz-Volterra projection onto finite element spaces and applications to integrodifferential and related equations. SIAM J.  Numer. Anal. 28 (1991), 1047–1070. MR 1111453
[29] G.  Marchuk, V.  Shaidurov: Difference Methods and Their Extrapolation. Springer, New York, 1983. MR 0705477
[30] B.  Neta, J.  Igwe: Finite difference versus finite elements for solving nonlinear integro-differential equations. J.  Math. Anal. Appl. 112 (1985), 607–618. MR 0813623
[31] A. K.  Pani, V.  Thomée, L. Wahlbin: Numerical methods for hyperbolic and parabolic integro-differential equations. J.  Integral Equations Appl. 4 (1992), 533–584. MR 1200801
[32] I. H.  Sloan, V.  Thomée: Time discretization of an integro-differential equation of parabolic type. SIAM J.  Numer. Anal. 23 (1986), 1052–1061. MR 0859017
[33] V.  Thomée, N.  Zhang: Error estimates for semidiscrete finite element methods for parabolic integro-differential equations. Math. Comput. 53 (1989), 121–139. MR 0969493
[34] J.  Wang: Superconvergence and extrapolation for mixed finite element methods on rectangular domains. Math. Comput. 56 (1991), 477–503. MR 1068807 | Zbl 0729.65084
[35] J.  Wang: Asymptotic expansions and $L^{\infty }$-error estimates for mixed finite element methods for second order elliptic problems. Numer. Math. 55 (1989), 401–430. MR 0997230
[36] N.  Yan, K.  Li: An extrapolation method for BEM. J.  Comput. Math. 2 (1989), 217–224. MR 1016842 | Zbl 0673.65072
[37] S.  Zhang, T.  Lin, Y.  Lin, M.  Rao: Extrapolation and a-posteriori error estimators of Petrov-Galerkin methods for non-linear Volterra integro-differential equations. J.  Comp. Math. 19 (2001), 407–422. MR 1842853
[38] A.  Zhou, C. B.  Liem, T. M.  Shih, T.  Lü: A multi-parameter splitting extrapolation and a parallel algorithm. Syst. Sci. Math. Sci. 10 (1997), 253–260. MR 1469184
[39] Q.  Zhu, Q.  Lin: Superconvergence Theory of the Finite Element Methods. Hunan Science Press, , 1989.
Partner of
EuDML logo