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(6) 




From: Mail Delivery System <MailerD<aemon@li...>  20061009 12:09:28

This message was created automatically by mail delivery software. A message that you sent has not yet been delivered to one or more of its recipients after more than 10 hours on the queue on externalmx1.sourceforge.net. The message identifier is: 1GWhbW0000e8IU The date of the message is: Sun, 08 Oct 2006 14:06:00 0200 The subject of the message is: force the Far section. The address to which the message has not yet been delivered is: libmeshusers@... Delay reason: SMTP error from remote mailer after RCPT TO:<libmeshusers@...>: host mail.sourceforge.net [66.35.250.206]: 451Could not complete sender verify callout 451Could not complete sender verify callout for 451<libmeshusers@...>. 451The mail server(s) for the domain may be temporarily unreachable, or 451they may be permanently unreachable from this server. In the latter case, 451you need to change the address or create an MX record for its domain 451if it is suppos No action is required on your part. Delivery attempts will continue for some time, and this warning may be repeated at intervals if the message remains undelivered. Eventually the mail delivery software will give up, and when that happens, the message will be returned to you. 
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 
From: Karl Tomlinson <k.tomlinson@au...>  20061009 03:44:41

On Fri, 6 Oct 2006 13:26:34 0500, Benjamin Kirk wrote: > The DenseMatrix and DenseVector condense() function implements > exactly what John says, and can be used if you know the degree > of freedom values on the boundary nodes. Thanks for this  I'll check these out. > For a noninterpolary basis you usually don't know the values a > priori, or at least they are not trivial to obtain. For that > reason you can add the penalty of the L2 projection of the > boundary data, which works in general. I can see that the penalty method can work in more cases, but there are still some cases where I can't see how the penalty method can work well. I can see that the penalty method works well if determining the values for the degrees of freedom involved in satisfying the Dirichlet boundary conditions can be considered separately from solving domain equations for the other degrees of freedom. 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. i.e. the boundary conditions remain satisfied provided the values of these degrees of freedom satisfy a nonfullrank system of equations. 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, then their values should be determined by the domain equation contributions, which are inconsistent and are trumped by the boundary projections. Choosing too small a penalty coefficient results in errors from the inconsistent equations, and it looks like choosing too large a coefficient would result in numerical errors due to the trumping of the domain terms. 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. 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. 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. The Nitsche method can also be used on interfaces between portions of the domain with nonmatching meshs, as analysed in R. Becker, P. Hansbo, and R. Stenberg's A finite element method for domain decomposition with nonmatching grids in Mathematical Modelling and Numerical Analysis 37 (2003) 209225 (http://www.math.hut.fi/~rstenber/Publications/BeckerHansboStenberg.pdf). 