Top

Journal of Applied Mathematics and Computing

Published in:

01-07-2013 | Original Research

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

Authors: Hui Guo, Huajie Wang

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2013

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Positive solutions of m-point boundary value problems for a class of nonlinear fractional differential equations

next article A posteriori error analysis for defect correction method for two parameter singular perturbation problems

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

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., Ginn, T.R.: Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Transp. Porous Media 13, 123–138 (1995) CrossRef

Dagan, G.: The significance of heterogeneity of evolving scales to transport in porous media. Water Resour. Res. 30, 3327–3336 (1994) CrossRef

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)

Gurtin, M., Pipkin, A.: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968) MathSciNetMATHCrossRef

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

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

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

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.

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.

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.

Guo, H.: Least-squares Galerkin procedures for parabolic integro-differential equations. Appl. Math. Comput. 150, 749–762 (2004) MathSciNetMATHCrossRef

13.

Guo, H.: Crank-Nicolson least-squares Galerkin procedures for parabolic integro-differential equations. Appl. Math. Comput. 180, 622–634 (2006) MathSciNetMATHCrossRef

14.

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.

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.

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.

Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, London (1969)

18.

Babuśka, I.: The finite element method with Lagrange multipliers. Numer. Math. 20, 179–192 (1973) MATHCrossRef

19.

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.

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.

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.

Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) MATH

23.

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

24.

Nedelec, J.C.: Mixed finite element in R ³. Numer. Math. 35, 315–341 (1980) MathSciNetMATHCrossRef

25.

Ciarlet, P.G.: Finite Element Methods for Elliptic Problems. North-Holland, New York (1978)

Title: Crank-Nicolson split least-squares procedure for nonlocal reactive flow in porous media
Authors: Hui Guo
Huajie Wang
Publication date: 01-07-2013
Publisher: Springer-Verlag
Published in: Journal of Applied Mathematics and Computing / Issue 1-2/2013
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-012-0627-z

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1-2/2013

Robust L ∞ reliable control for impulsive switched nonlinear systems with state delay

An iterative procedure for solving the common solution of two total quasi-ϕ-asymptotically nonexpansive multi-valued mappings in Banach spaces

Group inverse for two classes of 2×2 anti-triangular block matrices over right Ore domains

The recursion operator method for generalized symmetries and Bäcklund transformations of the Burgers’ equations

Blow up of series solutions

Asymptotic behavior of an SEIR epidemic model with quadratic treatment

Premium Partner