-----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 On 29/09/15 09:32, Justin Chang wrote:
Colin,
1) What do you mean by not supporting elimination within hybridized methods?
If you choose a dual formulation of your problem, an efficient means of solving it is to "hybridise" (breaking continuity between elements, followed by enforcing continuity with lagrange multipliers. One can then, in principle, eliminate all the variables element-wise onto the lagrange multipliers and solve a global system for that. An overview is given here: Bernardo Cockburn, Jayadeep Gopalakrishnan and Raytcho Lazarov. Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems, SIAM Journal on Numerical Analysis (2009)
2a) So if I use the interior penalty method that would be using DG1 space. Would projecting between DG1 and DG0 spaces result in a loss of accuracy and information?
2b) For the advection demo, I modified it so that it solves a transient problem, but when I use DG1 elements I seem to get answers that violate the maximum principles. Does the formulation in the demo only work for DG0 elements or could it also potentially work for DG1 elements?
I think it should work for DG1 elements. See for example the time-dependent advection tests in firedrake/tests/regression/test_periodic_interval_advection.py A guess as to why you might be getting maximum violation: Do you have steep gradients, shocks, or discontinuities in your solution? If so, then you may need to apply some form of slope limiting to avoid spurious oscillations. Dave Ketcheson has a nice ipython notebook running through some of the theory behind this (with pretty pictures!) http://nbviewer.ipython.org/gist/ketch/03da681c7966a8ce630f. Cheers, Lawrence -----BEGIN PGP SIGNATURE----- Version: GnuPG v1 iQEcBAEBAgAGBQJWClQkAAoJECOc1kQ8PEYvsWUH/3KUKLHh4EpzLyEr0inhyl5r ZXPrHkiWcFO1gtzzTrwAYg3gwUs8NIzqSrCLRC/pMGTEin94A0iumlE56ebDHa9c WPVjfH6GHwbij+fo4UfRMCFsiepM0eOq3oqeBBQb6ILxRenJYwqXrhNiPR3o1+Kq Bqa9pr2ZduitJwgcl9MstjGwXvYRdztZIQcbqFh+UeaKMED1toTxrBxMKicHXsC8 SQvXBIlhGomSjgZHrNrEY0JdDG8mIjcmtcB8DO17BnA4NpMEmohiR83LYE/PutPj YHXssmhilMNytzjE7lUR2x8Cbk4opVD1uohKNMZlVRnhr+dTA6hwETVyg1Sux80= =SRau -----END PGP SIGNATURE-----