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From: Roy Stogner <roystgnr@ic...>  20061010 15:31:47

On Tue, 10 Oct 2006, Karl Tomlinson wrote: > Roy Stogner writes: > >> 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. > > I'm pleased it still works fine. What size coefficient do you > use for the penalty term (and are the other terms of unit > magnitude)? Or does this not seem to matter much? It does matter, and I wish I had a better understanding of exactly how it matters. Make epsilon too large (e.g. 1e5), and your convergence soon bottoms out as approximation error is swamped by boundary error. Make epsilon too small (e.g. 1e15), and your convergence soon bottoms out as approximation error is swamped by floating point error. I generally set epsilon to 1e10 and cross my fingers. > How should the penalty parameter depend on mesh size? Good question. Obviously to get anything like a consistent formulation in exact arithmetic you probably need to decrease epsilon with h, and to avoid eventually overwhelming your finite element error you probably need some rate like h^{p+1}. But, in practice it seems like there's nothing wrong with using small epsilons on coarse meshes, and if you make epsilon too small on fine meshes then floating point error kills your solution. >> 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. > > I'm seeing a 1 in the numerator of the first term after the > summation sign and I think this is right. The equation seems to > behave appropriately in the limits of Remarks 24. I'm sorry, I meant to say the first term on the second line of equation 5... but on second glance that doesn't look like an obvious mistake either. I must have confused my q's and g's. >> Lemma 3 gives an upper bound, and equations 1415 suggest >> (perhaps misleadingly) that setting gamma too low will increase >> the final error. > > Looking at Remarks 57 in this paper and comparing Equation 32 in > http://math.tkk.fi/~rstenber/Publications/BeckerHansboStenberg.pdf, > I'm guessing that the gamma in equation 15 is a typo and should > not be there. (Warning: the gammas in the two papers seem to be > reciprocals.) > > This second paper goes into more detail on the bound for linear > elements but I haven't worked out why the bound seems to differ by > a factor of 4. I haven't read through the second paper yet; I'll see if I can figure it out today.  Roy 