From: David K. <dav...@gm...> - 2008-08-26 16:57:49
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> Well... yeah but it still feels it's a different class of > approximation deserving a different enum. Errors in computing the L2 > and H1 errors are due to quadrature error, which can be bounded in > terms of higher-order derivatives of the exact solution. The > approximate L_INF norm calculation (as we have defined it here) may > not have an error representation which is quite so well-defined ... > then again maybe it does? Seems to me it would depend strongly on the > number of sampling points as well. Yeah, I see what you mean. I suppose the ideal thing (I'm not saying this should be done in practice) would be to compute the interpolant of the error based on values at the quadrature points, and take the L_INFTY norm of the interpolant. Given a regularity assumption on the error, I'm sure there are bounds for the L_INFTY error of the interpolant. However, I think in some cases the maximum of the values at the interpolation points would be a good approximation to the supremum of the interpolant of the error. For example, if the interpolation points are Gauss quadrature points in 1D (or any points that are clustered like Chebyshev points), then I believe that the supremum of the polynomial interpolant will (asympotically) be very close to the maximum of the sampled values, and both of these would converge "spectrally" to the exact L_INFTY error. On the other hand, if we're using bad interpolation points, e.g. equally spaced points in 1D, then the supremum of the interpolant grows exponentially fast compared to the values at the interpolation points, so in that case the heuristic would fail horribly. Anyway, I guess what I'm saying is that I think you're right John, the quadrature point samples need not be a good approximation to the continuous L_INFTY norm, but perhaps it's OK as a heuristic...? >> Also, regarding the superconvergence issue, if we have superconvergence in >> the L_INF norm at the quadrature points, and we use that quadrature rule to >> compute the L2 error, then won't we just get the same superconvergence in >> the quadrature-based L2 error as well? > > I think you are right, so in general one should always use a different > quadrature rule, unless I am mistaken about that superconvergence > property. For the life of me, I can't remember where I heard that and > I'm starting to wonder if I may have made it up :-) It seems plausible to me. Or, at a minimum, I've definitely heard about superconvergence at the nodes of the mesh, and the user could use the nodes as quadrature points... - Dave |
From: Tim K. <tim...@ce...> - 2008-08-27 08:53:25
Attachments:
patch
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Dear John On Tue, 26 Aug 2008, John Peterson wrote: > Unfortunately the error is not a linear function in general, even > though the approximate solution may be. Yes. I was thinking about the case where the 'exact' solution is just a solution on a finer grid, in which case (if I understand the code correctly) the computation is performed on the fine grid and hence the 'error' is actually contained in the ansatz space. Of course, you could not know what I was thinking about. Please find attached the new patch that corrects the error I made in parallel for the L-infty norm. After all, I did not change the name of the L_INF norm since I think the result of the discussion is that this is the most sensible approximation and that it naturally corresponds to the approximation that is done for the other norms, too. Additionally, I added some comments that clarify this to the user. Best Regards, Tim -- Dr. Tim Kroeger Phone +49-421-218-7710 tim...@me..., tim...@ce... Fax +49-421-218-4236 MeVis Research GmbH, Universitaetsallee 29, 28359 Bremen, Germany Amtsgericht Bremen HRB 16222 Geschaeftsfuehrer: Prof. Dr. H.-O. Peitgen |
From: John P. <jwp...@gm...> - 2008-08-27 14:54:12
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Hi Tim, On Wed, Aug 27, 2008 at 3:53 AM, Tim Kroeger <tim...@ce...> wrote: > > Please find attached the new patch that corrects the error I made in > parallel for the L-infty norm. After all, I did not change the name of the > L_INF norm since I think the result of the discussion is that this is the > most sensible approximation and that it naturally corresponds to the > approximation that is done for the other norms, too. Additionally, I added > some comments that clarify this to the user. The patch has been checked in. -- John |