×
Vohralík, M.; Maryška, J.; Severýn, O.

Mixed and nonconforming finite element methods on a system of polygons. (English) Zbl 1112.65123

Appl. Numer. Math. 57, No. 2, 176-193 (2007).
The authors investigate the lowest-order Raviart-Thomas mixed finite element method for second-order elliptic problems posed over a system of intersecting two-dimensional polygons placed in three-dimensional Euclidian space. The theoretical results are finally verified on a model problem with a known analytical solution. The application of the proposed method to the simulation of a real problem is also discussed.
Reviewer: Răzvan Răducanu (Iaşi)

Cited in 12 Documents

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Keywords:

second-order elliptic problem; system of polygons; Raviart-Thomas mixed finite element method; nonconforming finite element method; fracture flow; numerical example
Cite Review PDF
Full Text: DOI

References:

[1] Adler, P. M., Fractures and Fracture Networks (1999), Kluwer Academic: Kluwer Academic Dordrecht
[2] Andersson, J.; Dverstorp, B., Conditional simulations of fluid flow in three-dimensional networks of discrete fractures, Water Resour. Res., 23, 1876-1886 (1987)
[3] Arnold, D. N.; Brezzi, F., Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér., 19, 7-32 (1985) · Zbl 0567.65078
[4] Bastian, P.; Chen, Z.; Ewing, R. E.; Helmig, R.; Jakobs, H.; Reichenberger, V., Numerical simulation of multiphase flow in fractured porous media, (Chen, Z.; Ewing, R. E.; Shi, Z. C., Numerical Treatmenet of Multiphase Flows in Porous Media (2000), Springer: Springer Berlin), 50-68 · Zbl 1072.76575
[5] Bear, J., Modeling flow and contaminant transport in fractured rocks, (Bear, J.; Tsang, C. F.; de Marsily, G., Flow and Contaminant Transport in Fractured Rock (1993), Academic Press: Academic Press New York), 1-38
[6] Bogdanov, I. I.; Mourzenko, V. V.; Thovert, J.-F.; Adler, P. M., Effective permeability of fractured porous media in insteady state flow, Water Resour. Res., 39, 1023 (2003)
[7] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer: Springer New York · Zbl 0788.73002
[8] Cacas, M. C.; Ledoux, E.; de Marsily, G.; Tillie, B.; Barbreau, A.; Durand, E.; Feuga, B.; Peaudecerf, P., Modeling fracture flow with a stochastic discrete fracture network: Calibration and validation 1. The flow model, Water Resour. Res., 26, 479-489 (1990)
[9] Chen, Z., Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems, East-West J. Numer. Math., 4, 1-33 (1996) · Zbl 0932.65126
[10] Dershowitz, W. S.; Fidelibus, C., Derivation of equivalent pipe network analogues for three-dimensional discrete fracture networks by the boundary element method, Water Resour. Res., 35, 2685-2691 (1999)
[11] Elsworth, D., A hybrid boundary element-finite element analysis procedure for fluid flow simulation in fractured rock masses, Int. J. Numer. Anal. Methods Geomech., 10, 569-584 (1986) · Zbl 0611.73104
[12] Hoteit, H.; Erhel, J.; Mosé, R.; Philippe, B.; Ackerer, Ph., Numerical reliability for mixed methods applied to flow problems in porous media, Comput. Geosci., 6, 161-194 (2002) · Zbl 1079.76581
[13] Koudina, N.; Gonzalez Garcia, R.; Thovert, J.-F.; Adler, P. M., Permeability of three-dimensional fracture networks, Phys. Rev. E, 57, 4466-4479 (1998)
[14] Long, J. C.S.; Gilmour, P.; Witherspoon, P. A., A model for steady state flow in random three dimensional networks of disc-shaped fractures, Water Resour. Res., 21, 1105-1115 (1985)
[15] G. Maros, K. Palotás, B. Koroknai, E. Sallay, G. Szongoth, Z. Kasza, L. Zilahi-Sebess, Core log evaluation of borehole Ptp-3 in the Krušné hory mts, MS Geological Institute of Hungary, Budapest, 2001; G. Maros, K. Palotás, B. Koroknai, E. Sallay, G. Szongoth, Z. Kasza, L. Zilahi-Sebess, Core log evaluation of borehole Ptp-3 in the Krušné hory mts, MS Geological Institute of Hungary, Budapest, 2001
[16] Maryška, J.; Severýn, O.; Vohralík, M., Mixed-hybrid FEM discrete fracture network model of the fracture flow, (Sloot, P. M.A.; etal., Proceedings of Computational Science—ICCS 2002 (2002), Springer: Springer Amsterdam), 794-803 · Zbl 1054.76051
[17] Maryška, J.; Severýn, O.; Vohralík, M., Numerical simulation of fracture flow with a mixed-hybrid FEM stochastic discrete fracture network model, Comput. Geosci., 8, 217-234 (2004) · Zbl 1116.76427
[18] Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1994), Springer: Springer Berlin · Zbl 0852.76051
[19] Raviart, P.-A.; Thomas, J.-M., A mixed finite element method for 2-nd order elliptic problems, (Galligani, I.; Magenes, E., Mathematical Aspects of Finite Element Methods. Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., vol. 606 (1977), Springer: Springer Berlin), 292-315 · Zbl 0362.65089
[20] Roberts, J. E.; Thomas, J.-M., Mixed and hybrid methods, (Ciarlet, P. G.; Lions, J.-L., Handbook of Numerical Analysis, vol. 2 (1991), Elsevier: Elsevier Amsterdam), 523-639 · Zbl 0875.65090
[21] Slough, K. J.; Sudicky, E. A.; Forsyth, P. A., Numerical simulations of multiphase flow and phase partitioning in discretely fractured geological media, J. Contam. Hydrol., 40, 107-136 (1999)
[22] J.-M. Thomas, Sur l’analyse numérique des méthodes d’Éléments finis hybrides et mixtes, Ph.D. Dissertation, Université Pierre et Marie Curie (Paris 6), 1977; J.-M. Thomas, Sur l’analyse numérique des méthodes d’Éléments finis hybrides et mixtes, Ph.D. Dissertation, Université Pierre et Marie Curie (Paris 6), 1977
[23] Vohralík, M.; Maryška, J.; Severýn, O., Mixed-hybrid discrete fracture network model, (Chen, Z.; Glowinski, R.; Li, K., Current Trends in Scientific Computing. Current Trends in Scientific Computing, Contemporary Mathematics, vol. 329 (2003), American Mathematical Society: American Mathematical Society Providence, RI), 325-332 · Zbl 1041.76046
[24] Wanfang, Z.; Wheater, H. S.; Johnston, P. M., State of the art of modelling two-phase flow in fractured rock, Envir. Geol., 31, 157-166 (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.