Mixed Discontinuous Galerkin Methods for Darcy Flow

Abstract

We consider a family of mixed finite element discretizations of the Darcy flow equations using totally discontinuous elements (both for the pressure and the flux variable). Instead of using a jump stabilization as it is usually done for discontinuous Galerkin (DG) methods (see e.g. D. N. Arnold et al. SIAM J. Numer. Anal. 39, 1749-1779 (2002) and B. Cockburn, G. E. Karniadakis and C.-W. Shu, DG methods: Theory, computation and applications, (Springer, Berlin, 2000) and the references therein) we use the stabilization introduced in A. Masud and T. J. R. Hughes, Meth. Appl. Mech. Eng. 191, 4341-4370 (2002) and T. J. R. Hughes, A. Masud, and J. Wan, (in preparation). We show that such stabilization works for discontinuous elements as well, provided both the pressure and the flux are approximated by local polynomials of degree ≥ 1, without any need for additional jump terms. Surprisingly enough, after the climination of the flux variable, the stabilization of A. Masud and T. J. R. Hughes, Meth. Appl. Mech. Eng 191, 4341-4370 (2002) and T. J. R. Hughes, A. Masud, and J. Wan, (in preparation) turns out to be in some cases a sort of jump stabilization itself, and in other cases a stable combination of two originally unstable DG methods (namely, Bassi-Rebay F. Bassi and S. Rebay, Proceedings of the Conference "Numerical methods for fluid dynamics V", Clarendon Press, Oxford (1995) and Baumann-Oden Comput. Meth. Appl. Mech. Eng 175, 311-341 (1999).