The effect of numerical integration in mixed finite element approximation in the simulation of miscible displacement

 
 
 
  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract


    We consider the effect of numerical integration in finite element  procedures applied to a nonlinear system of two coupled partial differential equations describing the miscible displacement of one incompressible fluid by another in a porous meduim. We consider the use of the numerical quadrature scheme for approximating the pressure and velocity by a mixed method using Raviart - Thomas space of index  and the concentration by a standard Galerkin method. We also give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order estimates are derived when the imposed external flows are smoothly distributed.


  • Keywords


    Mixed finite element; Raviart-Thomas spaces; Quadrature scheme; Molecular dispersion.

  • References


      [1] P.G. Ciarlet, The Finite Element Method for elliptic problems, North Holland, Amsterdam, 1978.

      [2] P.A. Raviart, The use of numerical integration in finite element methods for solving parabolic equations, in Topics in Numerical Analysis, J.H. Miller, Ed., Academic Press, New York, 1973

      [3] P.G. Ciarlet and P.A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, in the mathematical foundations of the Finite Element Method with application to Partial Differential Equations, A.K. Aziz, ED., Academic Press, New York, 1972.

      [4] So-Hsiang Chou and Li Qian, The effect of Numerical Integration in Finite Element Methods for Nonlinear Parabolic Equations,} Numerical Methods for Partial Differential Equations, 6, 263-274 (1990).

      [5] Li Qian, Wang Daoyu , The effect of Numerical Integration in Finite Element Methods for Nonlinear Hyperbolic Equations,} Pure and Applied Mathematics, 2(1991), 57-61.

      [6] J. Douglas Jr., R. E. Ewing and M. F. Wheeler, Approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numer. 17(1983) 17-33.

      [7] J. Douglas Jr., R. E. Ewing and M. F. Wheeler, A time discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO Anal. Numer. 17(1983) 249-265.

      [8] R.E. Ewing, T. F. Russell and M. F. Wheeler, Convergence analysis of an approxiamtion of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Computer methods in applied mechanics and engineering 47(1984) 73-92.

      [9] J. Douglas Jr. and J. E. Roberts, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comp., 1983 41: 441-459.

      [10] Yuan Yi-rang , Time stepping along characteristics for the finite element approxiamtion of compressible miscible displacement in porous media,} Math. Numer. Sinica, 14(4): 385-406 (1992).

      [11] Li Qian and Chou So-Hsiang , Mixed methods for compressible miscible displacement with the effect of molecular dispersion, Acta Mathematicae Applicatae Sinica, Apr., 1995, Vol. 11(2).

      [12] M. F. Wheeler, A priori L2-error estimates for Galerkin approximates to parabolic partial differential equations, SIAM. J. Numer. Anal. 10, 723-759(1973).

      [13] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arinsing from Lagrangian multipliers, R.A.I.R.O., Anal. Numer. 2(1974), 129-151.

      [14] P. A. Raviart and J. M. Thomas, A mixed finite element for 2nd order elliptic problems, Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics 606, Springer-Verlag, 1977.

      [15] Nguimbi Germain, The effect of Numerical Integration in Finite Element Methods for Nonlinear Parabolic Equations, Appl. Math. J.Chinese Univ. Ser. B, 2001, 16(2): 219-230.

      [16] Nguimbi Germain, The effect of Numerical Integration in Finite Element Methods for Nonlinear Parabolic Integrodifferential Equations,Shandong University (Natural Science) Journal of Shandong University. 2001, 36(1), 31-41.

      [17] Nguimbi Germain, The effect of Numerical Integration in Finite Element Methods for Nonlinear Sobolev Equations, Numerical Mathematics, Journal of Chinese Universities. 2000, 9(2): 222-233.


 

View

Download

Article ID: 7320
 
DOI: 10.14419/ijamr.v6i2.7320




Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.