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Mixed discontinuous Galerkin methods for Darcy flow. (English) Zbl 1103.76031
The authors consider a family of mixed finite element discretizations of Darcy flow equations using totally discontinuous elements (both for the pressure and flux variable). Instead of using a jump stabilization as it is usually done in discontinuous Galerkin methods, they use the stabilization introduced in {\it A. Masud} and {\it T. J. R. Hughes} [Comput. Methods Appl. Mech. Eng. 191, No. 39--40, 4341--4370 (2002; Zbl 1015.76047)]. They show that such stabilization works for discontinuous elements as well, provided that both the pressure and the flux are approximated by local polynomials of degree greater than 1, without any need for additional jump terms.

76M10Finite element methods (fluid mechanics)
76S05Flows in porous media; filtration; seepage
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N12Stability and convergence of numerical methods (BVP of PDE)
Full Text: DOI
[1] Agmon, S. (1965). Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathematical Studies, Princeton, NJ. · Zbl 0142.37401
[2] · Zbl 0482.65060 · doi:10.1137/0719052
[3] · Zbl 1008.65080 · doi:10.1137/S0036142901384162
[5] · Zbl 1030.76026 · doi:10.1002/fld.562
[6] · Zbl 0924.76051 · doi:10.1016/S0045-7825(98)00359-4
[7] · Zbl 0599.65072 · doi:10.1007/BF01389710
[8] Brezzi, F., Manzini, G., Marini, D., Pietra, P., and Russo, A. (1999). Discontinuous finite elements for diffusion problems. Atti Convegno in onore di F. Brioschi (Milano 1997), Istituto Lombardo, Accademia di Scienze e Lettere, pp. 197--217.
[9] · doi:10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y
[10] · Zbl 1070.65117 · doi:10.1142/S0218202504003866
[17] · Zbl 0622.76077 · doi:10.1016/0045-7825(86)90025-3
[18] Hughes, T. J. R., Masud, A., and Wan, J. A. stabilized mixed discontinuous Galerkin method for Darcy flow, in preparation. · Zbl 1120.76040
[19] · Zbl 1015.76047 · doi:10.1016/S0045-7825(02)00371-7
[21] · Zbl 0384.65058 · doi:10.1137/0715010