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Mixed-hybrid finite elements and streamline computation for the potential flow problem. (English) Zbl 0767.76029
The authors discussed the application of a mixed-hybrid finite element method for solving equations of the form $$u=-A\nabla\varphi$$, $$\nabla\cdot u=f$$, where $$A$$ is a symmetric and uniformly positive- definite second-rank tensor. The lowest-order mixed method is presented in detail. This equation is fundamental in the theory of heat conduction, electrostatics, and ground-water hydraulics.
The mixed finite element method results in a large system of linear equations. The choice of a numerical method to solve this system is restricted by the fact that its coefficient matrix is indefinite. This drawback can be circumvented by an implementation technique called hybridization, which leads to a sparse and symmetric positive-definite system of linear equations. This system can be solved efficiently by the preconditioned conjugate gradient method, where the preconditioning matrix is constructed by the incomplete Cholesky decomposition or the modified incomplete Cholesky decomposition. The applicability and advantages of the mixed finite element method and the efficient solution of the resulting system of linear equations are illustrated by several numerical experiments.
The benefits of the mixed method are apparent for problems with rough tensors of hydraulic conductivity and especially if the domain is subdivided into very flat subdomains. After an approximation, $$u$$ has been computed by the mixed finite element method; streamlines and residence times can be determined efficiently and accurately using elementwise computations at the element level.
Reviewer: C.T.Tsai

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 76S05 Flows in porous media; filtration; seepage 86A05 Hydrology, hydrography, oceanography
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##### References:
 [1] Hydraulics of Groundwater, McGraw-Hill, New York, 1979. [2] and , ”A mixed finite element method for 2nd-order elliptic problems,” in Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics Vol. 606, and , Eds., Springer, Berlin, 1977, pp. 292-315. [3] ”Sur I’Analyse Numérique des Méthodes d’Eléments Finis Hybrides et Mixtes,” Ph.D. thesis, University Pierre et Marie Curie, Paris, 1977. [4] and , ”Mixed and hybrid finite element methods,” in Handbook of Numerical Analysis, Volume II: Finite Element Methods, and , Eds., North-Holland, Amsterdam, 1991, pp. 523-639. · Zbl 0875.65090 [5] Nédélec, Numer. Math. 35 pp 315– (1980) [6] Nédélec, Numer. Math. 50 pp 57– (1986) [7] , , and , ”Mixed finite element methods for accurate fluid velocities,” in Finite Elements in Fluids, Volume 6: Finite Elements and Flow Problems, , and , Eds., Wiley, New York, 1985, pp. 233-249. [8] ”Een Hybride Gemengde Eindige-Elementenmethode met Postprocessing, Toegepast op het Hall-Probleem”, Masters thesis, Eindhoven University of Technology, 1987. [9] and , Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986. [10] Arnold, Math. Modell. Numer. Ana. 19 pp 7– (1985) [11] and , Matrix Computations, North Oxford Academic, Oxford, 1983. [12] The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [13] Theorie und Numerik elliptischer Differentialgleichungen, Teubner, Stuttgart, 1986. · Zbl 0609.65065 [14] Brezzi, Rev. Fr. Autom. Inf. Rech. Opér 8-R2 pp 129– (1974) [15] and , Methods of Theoretical Physics, McGraw-Hill, New York, 1953. [16] The Finite Element Method, Academic, New York, 1988. [17] ”Mixed finite elements for the water flooding problems,” in Numerical Methods for Coupled Problems, , and , Eds., Pineridge, Swansea, 1981, pp. 968-976. [18] Hecht, RAIRO Anal. Numér. 15 pp 119– (1981) [19] Marini, SIAM J. Numer. Anal. 22 pp 493– (1985) [20] and , Finite Elements, Volume II: A Second Course, Prentice-Hall, Englewood Cliffs, 1983. [21] Weiser, SIAM J. Numer. Anal. 25 pp 351– (1988) [22] and , Finite Element Solution of Boundary Value Problems, Academic, New York, 1984. [23] Axelsson, BIT 25 pp 166– (1985) [24] Concus, SIAM J. Sci. Stat. Comput. 6 pp 220– (1985) [25] Meijerink, Math. Comput. 31 pp 148– (1977) [26] Meijerink, J. Comput. Phys. 44 pp 131– (1981) [27] ”Modified incomplete Cholesky (MIC) methods,” in Preconditioning Methods, Theory and Applications, Ed., Gordon and Breach, New York, 1983, pp. 265-293. · Zbl 0767.65017 [28] Kaasschieter, BIT 29 pp 824– (1989) [29] Gustafsson, Comput. Methods Appl. Mech. Eng. 55 pp 201– (1986) [30] Hughes, Comput. Methods Appl. Mech. Eng. 36 pp 241– (1983) [31] Chavent, Comput. Methods Appl. Mech. Eng. 47 pp 93– (1984) [32] Kaasschieter, BIT 28 pp 308– (1988) [33] Tóth, J. Geophy. Res. 68 pp 4795– (1963) [34] Physical Principles of Oil Production, McGraw-Hill, New York, 1949. [35] Philip, Transp. Porous Media 1 pp 319– (1986) [36] ”The convergence behaviour of preconditioned CG and CG-S.” in Preconditioned Conjugate Gradient Methods, Lecture Notes in Mathematics Vol. 1457, and , Eds., Springer, Berlin, 1990, pp. 126-136.
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