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Two-Grid Method for Miscible Displacement Problem by Mixed Finite Element Methods and Mixed Finite Element Method of Characteristics

Part of: Equations and systems of special type Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 May 2016

Yanping Chen  and
Hanzhang Hu
Yanping Chen*
Affiliation:
School of Mathematical Science, South China Normal University, Guangzhou 520631, Guangdong, P.R. China
Hanzhang Hu*
Affiliation:
School of Mathematical Science, South China Normal University, Guangzhou 520631, Guangdong, P.R. China School of Mathematics, Jiaying University, Meizhou 514015, Guangdong, P.R. China
*
*Corresponding author. Email addresses:[email protected](Y. Chen), [email protected](H. Hu)
*Corresponding author. Email addresses:[email protected](Y. Chen), [email protected](H. Hu)
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Abstract

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the approximation of the pressure equation and mixed finite element method with characteristics for the concentration equation. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this full discrete scheme problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = H2 in this paper. Finally, numerical experiment indicates that two-grid algorithm is very effective.

Keywords

Two-grid methodmiscible displacement problemmixed finite elementcharacteristic finite element method

MSC classification

Secondary: 35M13: Initial-boundary value problems for equations of mixed type 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1503 - 1528
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Brezzi, F.. On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. RAIRO: Anal. Numer., 2:129–151,1974.Google Scholar
[2]Chen, L. and Chen, Y.. Two-grid discretization scheme for nonlinear reaction diffusion equations by mixed finite element methods. Advances in Applied Mathematics and Mechanics, 6:203–219,2014.Google Scholar
[3]Chen, Y., Huang, Y., and Yu, D.. A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Int. J. Numer. Mech. Engng, 57:193–209, 2003.Google Scholar
[4]Chen, Y. and Li, L.. Lp error estimates of two-grid schemes of expanded mixed finite element methods. Applied Mathematics and Computation, 209:197–205,2009.CrossRefGoogle Scholar
[5]Chen, Y., Liu, H., and Liu, S.. Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods. Internat. J. Numer.Methods in Engineering, 69:408–422, 2007.Google Scholar
[6]Chen, Z.. Expanded mixed element methods for linear second-order elliptic problems(i). RAIRO: Math. Model. Numer. Anal., 32:479–499,1998.Google Scholar
[7]Chen, Z. and Ewing, R.. Mathmatical analysis for reservior models. SIAM Journal on Mathematic al Analysis, 30(2):431–453,1999.CrossRefGoogle Scholar
[8]Dawson, C., Russell, T., and Wheeler, M.. Some improved error estimates for the modified method of characteristics. SIAMJ. Numer. Anal., 26:1487–1512,1989.Google Scholar
[9]Dawson, C. N., Wheeler, M. F., and Woodward, C. S.. A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal., 35:435–452,1998.Google Scholar
[10]Ewing, E. and Wheeler, M. F.. Galerkin methods for miscible displacement problems in porous media. SIAM Journal on Numerical Analysis, 17:351–365,1980.Google Scholar
[11]Ewing, R. E., Russell, T. F., and Wheeler, M. F.. Simulation of miscible displacement using mixed methods and a modified method of characteristics. Paper SPE 12241, Proceedings, Sev¬enth SPE Symposium on Reservoir Simulation, Society of petroleum Engineers, Dallas, TX, pages 71-81,1983.Google Scholar
[12]Ewing, R. E., Russell, T. F., and Wheeler, M. F.. Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Computer Methods in Applied Mechanics and Engineering, 47:73–92,1984.Google Scholar
[13]Jin, J., Shu, S., and Xu, J.. A two-grid discretization method for decoupling systems of partial differential equations. Mathematics of Computation, 75:1617–1626,2006.Google Scholar
[14]J. D., Jr. Simulation of miscible displacement in porous media by a modified method of characteristic procedure. Numerical analysis, 1981.Google Scholar
[15]J. D., Jr.Finite difference methods for two-phase incompressible flow in porous media. SIAM J. Numer. Anal., 20:681–696,1983.Google Scholar
[16]J. D., Jr., Ewing, R., and Wheeler, M.F.. The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO: Analyse Numerique, 17:17–33,1983.Google Scholar
[17]J. D., Jr., Ewing, R. E., and Wheeler, M.. A time-discretization procedure for a mixed finite elememt approximation of miscible displacement in porous media. RAIRO: Analyse Numerique Numerical Analysis, 17:249–265,1983.Google Scholar
[18]J. D., Jr. and Roberts, J. E.. Mixed finite element methods for second order elliptic problems. Mat. Apl. Comput., 1:91–103,1982.Google Scholar
[19]J. D., Jr. and Roberts, J. E.. Global estimates for mixed finite element methods for second order elliptic equations. Math. Comput., 44:39–52,1985.Google Scholar
[20]J. D., Jr. and Russell, T. F.. Numerical methods for convention-dominated diffusion problems base on combining the method of characteristics with finite element or finite difference procedures. SIAMJ. Numer. Anal., 19(5):871–885,10 1982.Google Scholar
[21]Xu, J.. A novel two-grid method for semilinear equations. SIAM Journal on Scientific Computing, 15:231–237,1994.CrossRefGoogle Scholar
[22]Xu, J.. Two-grid discretization techniques for linear and non-linear pdes. SIAM Journal on Numerical Analysis, 33:1759–1777,1996.Google Scholar
[23]Yuan, Y.. Characteristic finite element methods for positive semidefinite problem of two phase miscible flow in three dimensions. Sci. Bull. China, 22:2027–2032,1996.Google Scholar
[24]Yuan, Y.. Characteristic finite difference methods for positive semidefinite problem of two phase miscible flow in porous media. Syst. Sci. Math. Sci. China, 4:299–306,1999.Google Scholar
[25]Zaglmayr, S.. High order finite element methods for electromagnetic field computation. na, 2006.Google Scholar
[26]Zhou, J., Chen, L., Huang, Y., and Wang, W.. An efficient two-grid scheme for the cahn-hilliard equation. Commun. Computat. Phys., 17(1):127–145,2015.Google Scholar