Abstract
A Schwarz type domain decomposition algorithm is formulated to solve an approximation for miscible displacement of compressible fluids in porous media and convergence rate of the algorithm is analyzed. First, a splitting positive definite mixed element procedure is used to treat the pressure equation of parabolic type. The coefficient matrix of the mixed element system is symmetric positive definite and the flux equation is separated from the pressure equation so that the approximate solution of the flux function can be independently obtained, and a characteristic finite element method is used to treat the convectiondiffusion equations of the concentrations. Then a Schwarz type domain decomposition algorithm is introduced to solve the symmetric positive definite systems step by step. How many iterative cycles are needed at each time level? A convergence analysis is given.
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References
Adams, R. A., Sobolev Spaces, New York. Academic Press, 1975.
Brezzi, F., Douglas, J., Jr., Duran, R., and Marini, L. D., Mixed finite elements for second order elliptic problems in three space variables, Numer. Math. 51 (1987), 237–250.
Brezzi, F., Douglas, J., Jr., Fortin, M., and Marini, L. D., Efficient rectangular mixed finite elements in two and three space variables, RAIRO Model Math. Anal. Numer. 4(21) (1987), 581–604.
Brezzi, F., Douglas, J., Jr., and Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), 217–235.
Chen, Z. and Douglas, J., Jr., Prismatic mixed finite elements for second order elliptic problems, Calcolo 26 (1989), 135–148.
Chou, S. H. and Li, Q., Mixed finite element methods for compressible miscible displacement in porous media, Math. Comp. 57 (1991), 507–527.
Ciarlet, P. G., The Finite Element Methods for Elliptic Problems, North-Holland, New York, 1978.
Douglas, J., Jr., The numerical simulation of miscible displacement in porous media, Comput. Methods Nonlin. Mech., North-Holland, Amsterdam, 1980.
Douglas, J., Jr., Finite difference methods for two-phase incompressible flow in porous media, SIAM J. Numer. Anal. 20 (1983), 681–696.
Douglas, J., Jr. and Roberts, J. E., Numerical methods for a model for compressible miscible Displacement in porous Media, Math. Comp. 41 (1983), 441–459.
Douglas, J., Jr., Simulation of miscible displacement in porous media by a modified method of characteristics procedure, Numerical Analysis. Lecture Notes in Math 912, Springer-Verlag, Berlin, 1982.
Douglas, J., Jr., Ewing, R. E., and Wheeler, M. F., The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO, Anal. Numer. 17 (1983), 17–33.
Douglas, J., Jr., Ewing, R. E., and Wheeler, M. F., A time-discretization procedure for a mixed finite element approximation of miscible displacement in poro media, RAIRO Anal. Numer. 17 (1983), 249–256.
Duran, R., On the approximation of miscible displacement in porous media by the modified methods of characteristics combined with a mixed method. SIAM J. Numer. Anal. 25 (1988), 989–1001.
Dryja, M. and Widlund, O. B., Some domain decomposition algorithms for elliptic problems, Proceedings of the Conference on Iterative Methods for Large Linear System held in Austin, D. Young, Jr. ed., Academic Press, Orlando, 1989, 121–134.
Lions, P. L., On Schwarz alternating method, Part I, Proceedings of the first international symposium on domain decomposition methods for partial differential equations, SIAM. Philadelphia, Glowinski, R., Golub, G. H., Meurant, G. A., and Périaux, J. (eds.), 1988.
Lions, P. L., On Schwarz alternating method, Part II, Proceedings of the second international symposium on domain decomposition methods for partial differential equations, SIAM. Philadelphia, Chan, T., Glowinski, R., Périaux, J., and Widlund, O. B., eds., 1989.
Raviart, P. A. and Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math 606, Springer-Verlag, Berlin and New York, 1977, 292–315.
Wheeler, M. F., A priori error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 4 (1973), 723–759.
Yang, D. P., Approximation and its optimal error estimate of displacement of two-phase incompressible flow by mixed finite element and a modified method of characteristics, Chinese Science Bulletin 35 (1990), 1686–1689.
Yang, D. P., A characteristic mixed finite element method for displacement problem of compressible flows in porous media, Science in China, Series A (1998).
Yuan, Y. R., Characteristic-mixed finite element method for enhanced oil recovery simulation and optimal order error estimate, Chinese Science Bulletin 38 (1993), 1761–1766.
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Yang, D. (2000). Domain Decomposition Algprithm for a New Characteristic Mixed Finite Element Method for Compressible Miscible Displacement. In: Chen, Z., Ewing, R.E., Shi, ZC. (eds) Numerical Treatment of Multiphase Flows in Porous Media. Lecture Notes in Physics, vol 552. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45467-5_31
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DOI: https://doi.org/10.1007/3-540-45467-5_31
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