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A non-mortar mixed finite element method for elliptic problems on non-matching multiblock grids. (English) Zbl 0923.76100

Summary: We consider the approximation of second-order elliptic equations on domains that can be described as a union of sub-domains or blocks. We assume that a grid is defined on each block independently, so that the resulting grid over the entire domain need not be conforming (i.e. match) across the block boundaries. Several techniques have been developed to approximate elliptic equations on multiblock grids that utilize a mortar finite element space defined on the block boundary interface itself. We define a mixed finite element method that does not use such a mortar space. The method has an advantage in the case where adaptive local refinement techniques will be used, in that there is no mortar grid to refine. As is typical of mixed methods, our method is locally conservative element-by-element; it is also globally conservative across the block boundaries. Theoretical results show that the approximate solution converges at the optimal rate to the true solution. We present computational results to illustrate and confirm the theory.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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[1] T. Arbogast, L.C. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on non-matching multiblock grids, to appear.; T. Arbogast, L.C. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on non-matching multiblock grids, to appear. · Zbl 1001.65126
[2] Arbogast, T.; Dawson, C. N.; Keenan, P. T.; Wheeler, M. F.; Yotov, I., Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput., 18 (1997), to appear
[3] Arbogast, T.; Wheeler, M. F.; Yotov, I., Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal., 34, 828-852 (1997) · Zbl 0880.65084
[4] Arnold, D. N.; Brezzi, F., Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modèl, Math. Anal. Numèr., 19, 7-32 (1985) · Zbl 0567.65078
[5] Bernardi, C.; Maday, Y.; Patera, A. T., A new nonconforming approach to domain decomposition: the mortar element method, (Brezis, H.; Lions, J. L., Nonlinear Partial Differential Equations and Their Applications (1994), Longman Scientific & Technical: Longman Scientific & Technical UK) · Zbl 0797.65094
[6] Brezzi, F.; Douglas, J.; Duràn, R.; Fortin, M., Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51, 237-250 (1987) · Zbl 0631.65107
[7] Brezzi, F.; Douglas, J.; Fortin, M.; Marini, L. D., Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modèl. Math. Anal. Numèr., 21, 581-604 (1987) · Zbl 0689.65065
[8] Brezzi, F.; Douglas, J.; Marini, L. D., Two families of mixed elements for second order elliptic problems, Numer. Math., 88, 217-235 (1985) · Zbl 0599.65072
[9] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0788.73002
[10] Chen, Z.; Douglas, J., Prismatic mixed finite elements for second order elliptic problems, Calcolo, 26, 135-148 (1989) · Zbl 0711.65089
[11] Douglas, J.; Roberts, J. E., Global estimates for mixed methods for second order elliptic equations, Math. Comput., 44, 39-52 (1985) · Zbl 0624.65109
[12] Durán, R., Superconvergence for rectangular mixed finite elements, Numer. Math., 58, 287-298 (1990) · Zbl 0691.65076
[13] Ewing, R. E.; Lazarov, R. D.; Russell, T. F.; Vassilevski, P. S., Analysis of the mixed finite element method for rectangular Raviart-Thomas elements with local refinement, (Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (1990), SIAM: SIAM Philadelphia), 98-114 · Zbl 0705.65080
[14] Ewing, R. E.; Lazarov, R. D.; Wang, J., Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal., 28, 1015-1029 (1991) · Zbl 0733.65065
[15] Ewing, R. E.; Wang, J., Analysis of mixed finite element methods on locally refined grids, Numer. Math., 63, 183-194 (1992) · Zbl 0772.65071
[16] Glowinski, R.; Wheeler, M. F., Domain decomposition and mixed finite element methods for elliptic problems, (Glowinski, R.; etal., First International Symposium on Domain Decomposition Methods for Partial Differential Equations (1988), SIAM: SIAM Philadelphia), 144-172
[17] Nakata, M.; Weiser, A.; Wheeler, M. F., Some superconvergence results for mixed finite element methods for elliptic problems on rectangular domains, (Whiteman, J. R., The Mathematics of Finite Elements and Applications V (1985), Academic Press: Academic Press London), 367-389
[18] Nedelec, J. C., Mixed finite elements in \(R^3\), Numer. Math., 35, 315-341 (1980) · Zbl 0419.65069
[19] Raviart, R. A.; Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, (Mathematical Aspects of the Finite Element Method. Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, Vol. 606 (1977), Springer-Verlag: Springer-Verlag New York), 292-315
[20] Roberts, J. E.; Thomas, J.-M., Mixed and hybrid methods, (Ciarlet, P. G.; Lions, J., Handbook of Numerical Analysis, Vol. II (1991), Elsevier Science Publishers B.V.), 523-639 · Zbl 0875.65090
[21] Thomas, J. M., Sur l’analyse numerique des methodes d’elements finis hybrides et mixtes, (Thèse de Doctorat d’etat (1977), Sciences Mathematiques’a l’Universite Pierre et Marie Curie)
[22] Weiser, A.; Wheeler, M. F., On convergence of block-centered finite-differences for elliptic problems, SIAM J. Numer. Anal., 25, 351-375 (1988) · Zbl 0644.65062
[23] Yotov, I., Mixed finite element methods for flow in porous media, (Ph.D. Thesis (1996), Rice University: Rice University Houston, Texas) · Zbl 0897.76057
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