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From: David Knezevic <dave.knez@gm...>  20080826 16:57:49

> 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 higherorder 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 welldefined ... > 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 quadraturebased 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 