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Erschienen in: Journal of Applied Mathematics and Computing 1-2/2013

01.07.2013 | Original Research

Crank-Nicolson split least-squares procedure for nonlocal reactive flow in porous media

verfasst von: Hui Guo, Huajie Wang

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2013

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Abstract

In this paper, we introduce a Crank-Nicolson split least-squares Galerkin finite element procedure for parabolic integro-differential equations, arising in the modeling of nonlocal reactive flows in porous media. By selecting the least-squares functional properly, the procedure can be split into two independent sub-procedures, one of which is for the primitive unknown and the other is for the flux. By carefully choosing projections, we get optimal order H 1(Ω) and L 2(Ω) norm error estimates for u and sub-optimal (L 2(Ω)) d norm error estimate for σ with second-order accuracy in time increment. The numerical examples are given to testify the efficiency of the introduced scheme.

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Literatur
1.
Zurück zum Zitat Cushman, J.H., Hu, X., Deng, F.: Nonlocal reactive transfer with physical and chemical heterogeneity. Water Resour. Res. 31, 2219–2237 (1995) CrossRef Cushman, J.H., Hu, X., Deng, F.: Nonlocal reactive transfer with physical and chemical heterogeneity. Water Resour. Res. 31, 2219–2237 (1995) CrossRef
2.
Zurück zum Zitat Cushman, J.H., Ginn, T.R.: Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Transp. Porous Media 13, 123–138 (1995) CrossRef Cushman, J.H., Ginn, T.R.: Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Transp. Porous Media 13, 123–138 (1995) CrossRef
3.
Zurück zum Zitat Dagan, G.: The significance of heterogeneity of evolving scales to transport in porous media. Water Resour. Res. 30, 3327–3336 (1994) CrossRef Dagan, G.: The significance of heterogeneity of evolving scales to transport in porous media. Water Resour. Res. 30, 3327–3336 (1994) CrossRef
4.
Zurück zum Zitat Shelukin, V.V.: A nonlocal in time model for radionuclides propagation in stokes fluids dynamics of fluids with free boundaries. Din. Splosh. Sredy 107, 180–193 (1993) Shelukin, V.V.: A nonlocal in time model for radionuclides propagation in stokes fluids dynamics of fluids with free boundaries. Din. Splosh. Sredy 107, 180–193 (1993)
5.
Zurück zum Zitat Gurtin, M., Pipkin, A.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968) MathSciNetMATHCrossRef Gurtin, M., Pipkin, A.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968) MathSciNetMATHCrossRef
6.
Zurück zum Zitat Allegretto, W., Lin, Y., Zhou, A.: A box scheme for coupled systems resulting from microsensor thermistor problems. Dyn. Contin. Discrete Impuls. Syst. 5, 209–223 (1995) MathSciNet Allegretto, W., Lin, Y., Zhou, A.: A box scheme for coupled systems resulting from microsensor thermistor problems. Dyn. Contin. Discrete Impuls. Syst. 5, 209–223 (1995) MathSciNet
7.
Zurück zum Zitat Ewing, R.E., Lazarov, R., Lin, Y.: Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Methods Partial Differ. Equ. 16, 285–311 (2000) MathSciNetMATHCrossRef Ewing, R.E., Lazarov, R., Lin, Y.: Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Methods Partial Differ. Equ. 16, 285–311 (2000) MathSciNetMATHCrossRef
8.
Zurück zum Zitat Lin, Y., Thomas, V., Wahlbin, L.: Ritz-Volterra projection onto finite element spaces and application to integro-differential and related equations. SIAM J. Numer. Anal. 28, 1047–1070 (1995) CrossRef Lin, Y., Thomas, V., Wahlbin, L.: Ritz-Volterra projection onto finite element spaces and application to integro-differential and related equations. SIAM J. Numer. Anal. 28, 1047–1070 (1995) CrossRef
9.
Zurück zum Zitat Jiang, Z.: L ∞(L 2) and L ∞(L ∞) error estimates for mixed methods for integro-differential equations of parabolic type. Math. Model. Numer. Anal. 33, 531–546 (1999) MATHCrossRef Jiang, Z.: L (L 2) and L (L ) error estimates for mixed methods for integro-differential equations of parabolic type. Math. Model. Numer. Anal. 33, 531–546 (1999) MATHCrossRef
10.
Zurück zum Zitat Ewing, R.E., Lin, Y., Sun, T., Wang, J., Zhang, S.: Sharp L 2-error estimates and superconvergence of mixed finite element methods for non-Fichian flows in porous media. SIAM J. Numer. Anal. 40, 1538–1560 (2002) MathSciNetMATHCrossRef Ewing, R.E., Lin, Y., Sun, T., Wang, J., Zhang, S.: Sharp L 2-error estimates and superconvergence of mixed finite element methods for non-Fichian flows in porous media. SIAM J. Numer. Anal. 40, 1538–1560 (2002) MathSciNetMATHCrossRef
11.
Zurück zum Zitat Ewing, R.E., Lin, Y., Wang, J., Zhang, S.: L ∞-error estimates and superconvergence in maximum norm of mixed finite element methods for non-Fichian flows in porous media. Int. J. Numer. Anal. Model. 2, 301–328 (2005) MathSciNetMATH Ewing, R.E., Lin, Y., Wang, J., Zhang, S.: L -error estimates and superconvergence in maximum norm of mixed finite element methods for non-Fichian flows in porous media. Int. J. Numer. Anal. Model. 2, 301–328 (2005) MathSciNetMATH
12.
Zurück zum Zitat Guo, H.: Least-squares Galerkin procedures for parabolic integro-differential equations. Appl. Math. Comput. 150, 749–762 (2004) MathSciNetMATHCrossRef Guo, H.: Least-squares Galerkin procedures for parabolic integro-differential equations. Appl. Math. Comput. 150, 749–762 (2004) MathSciNetMATHCrossRef
13.
Zurück zum Zitat Guo, H.: Crank-Nicolson least-squares Galerkin procedures for parabolic integro-differential equations. Appl. Math. Comput. 180, 622–634 (2006) MathSciNetMATHCrossRef Guo, H.: Crank-Nicolson least-squares Galerkin procedures for parabolic integro-differential equations. Appl. Math. Comput. 180, 622–634 (2006) MathSciNetMATHCrossRef
14.
Zurück zum Zitat Cai, Z., Lazarov, R., Manteuffel, T.A., Miccormick, S.F.: First-order least squares for second-order partial differential equations, Part I. SIAM J. Numer. Anal. 31, 1785–1799 (1994) MathSciNetMATHCrossRef Cai, Z., Lazarov, R., Manteuffel, T.A., Miccormick, S.F.: First-order least squares for second-order partial differential equations, Part I. SIAM J. Numer. Anal. 31, 1785–1799 (1994) MathSciNetMATHCrossRef
15.
Zurück zum Zitat Pehlivanov, A., Carey, G.F., Lazarov, D.: Least-squares mixed finite elements for second-order elliptic problems. SIAM J. Numer. Anal. 31(5), 1368–1377 (1994) MathSciNetMATHCrossRef Pehlivanov, A., Carey, G.F., Lazarov, D.: Least-squares mixed finite elements for second-order elliptic problems. SIAM J. Numer. Anal. 31(5), 1368–1377 (1994) MathSciNetMATHCrossRef
16.
Zurück zum Zitat Yang, D.P.: Some least-squares Galerkin procedures for first-order time-dependent convection-diffusion system. Comput. Methods Appl. Mech. Eng. 180, 81–95 (1999) MATHCrossRef Yang, D.P.: Some least-squares Galerkin procedures for first-order time-dependent convection-diffusion system. Comput. Methods Appl. Mech. Eng. 180, 81–95 (1999) MATHCrossRef
17.
Zurück zum Zitat Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, London (1969) Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, London (1969)
18.
Zurück zum Zitat Babuśka, I.: The finite element method with Lagrange multipliers. Numer. Math. 20, 179–192 (1973) MATHCrossRef Babuśka, I.: The finite element method with Lagrange multipliers. Numer. Math. 20, 179–192 (1973) MATHCrossRef
19.
Zurück zum Zitat Brezzi, F.: On the existence uniqueness and approximation of saddle point problems arising from Lagrange multipliers. RAIRO. Anal. Numér. 8, 129–151 (1974) MathSciNetMATH Brezzi, F.: On the existence uniqueness and approximation of saddle point problems arising from Lagrange multipliers. RAIRO. Anal. Numér. 8, 129–151 (1974) MathSciNetMATH
20.
Zurück zum Zitat Rui, H.X., Kim, D., Kim, K.D.: A remark on least-squares mixed element methods for reaction-diffusion problems. J. Comput. Appl. Math. 202(2), 230–236 (2007) MathSciNetMATHCrossRef Rui, H.X., Kim, D., Kim, K.D.: A remark on least-squares mixed element methods for reaction-diffusion problems. J. Comput. Appl. Math. 202(2), 230–236 (2007) MathSciNetMATHCrossRef
21.
Zurück zum Zitat Rui, H.X., Kim, K.D., Kim, D.: Split least-squares finite element methods for linear and nonlinear parabolic problems. J. Comput. Appl. Math. 223, 938–952 (2009) MathSciNetMATHCrossRef Rui, H.X., Kim, K.D., Kim, D.: Split least-squares finite element methods for linear and nonlinear parabolic problems. J. Comput. Appl. Math. 223, 938–952 (2009) MathSciNetMATHCrossRef
22.
Zurück zum Zitat Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) MATH Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) MATH
23.
Zurück zum Zitat Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977) CrossRef Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977) CrossRef
25.
Zurück zum Zitat Ciarlet, P.G.: Finite Element Methods for Elliptic Problems. North-Holland, New York (1978) Ciarlet, P.G.: Finite Element Methods for Elliptic Problems. North-Holland, New York (1978)
Metadaten
Titel
Crank-Nicolson split least-squares procedure for nonlocal reactive flow in porous media
verfasst von
Hui Guo
Huajie Wang
Publikationsdatum
01.07.2013
Verlag
Springer-Verlag
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2013
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-012-0627-z

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