From: Roy Stogner <roystgnr@ic...>  20061009 05:38:34

On Mon, 9 Oct 2006, Karl Tomlinson wrote: > The situation seems different, however, if the degrees of freedom > that influence the values on the Dirichlet boundary are not > completely constrained by the boundary conditions. This is actually the case for some of the problems I've run. When using CloughTocher elements for second order problems, for example, in general it's only weighted sums of nodal gradient degrees of freedom that are constrained, but applying the penalty method on edge integrals still works fine. > For problems with natural boundary conditions, the equations > corresponding to degrees of freedom that influence the values on > the Dirichlet boundary condition will usually be inconsistent (not > satisfied by the exact solution). This is not a problem if the > penalty coefficient can be made so large that the L2 projection of > boundary data "trumps" the other contributions to the equations. > > However, if the boundary data projections don't completely > constrain the associated degrees of freedom, Could you give a concerete example where this wouldn't occur? I don't see even in theory how adding a heavily weighted ((ug),v) integral on the Dirichlet boundary edges wouldn't suffice, assuming that you're happy with solving the problem with Robin boundary conditions rather than Dirichlet. > The Nitsche method for Dirichlet boundary conditions looks like it > provides an attractive alternative. It is similar to the penalty > method but corrects the domain equations so that they are > consistent. That certainly sounds preferable. > There is still a coefficient to be selected for the Dirichlet > terms that depends on the mesh (for a positive definite system), > but it does not need to be so large as to swamp the domain > equations and so the system is better conditioned. As does that. > More details are in M. Juntunen and R. Stenberg's A finite element > method for general boundary conditions for the Proceedings of the > 18 Nordic Seminar on Computational Mechanics > (http://math.tkk.fi/~rstenber/Publications/nscm_general_boundary.pdf), > which also points out the inconsistency of the penalty method. Is there a typo in this paper? In the first term of equation 5, I would expect there to be a 1 in the numerator rather than an epsilon. How do you choose gamma in practice? Lemma 3 gives an upper bound, and equations 1415 suggest (perhaps misleadingly) that setting gamma too low will increase the final error. This looks interesting. I'll need to read it through again after I've had some sleep, though.  Roy 