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As motivation, consider a function which is smooth and continuous but for some reason it is very expensive to perform routine calculations of finding the Laplacian on it (maybe because it is over a large domain, etc.).

Given an original function, say, in one dimension, is it possible to estimate (or slightly overestimate) the maximum absolute value (i.e., the magnitude) of the Laplacian of that function anywhere over its given domain (without actually computing the Laplacian at all points and just finding the maximum)?

To start discussion, perhaps this can be done my taking a subset of points along the known function and using them as a basis for the estimate.

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Do you want the maximum eigenvalue (oscillatory and mesh dependent) of the differential operator or the inverse operator (the physically meaningful one). The differential operator is local so it should be inexpensive to evaluate. How much setup work are you willing to do? – Jed Brown Jun 1 '12 at 17:32
Do you have an analytical formula for the Laplacian of the given function? If so, interval arithmetic could be an option. – Geoff Oxberry♦ Jun 1 '12 at 18:45
I think what the OP is thinking of is more an analytic formula like the ones you get in the theory of complex variables: where, for example, you can compute the strength of a singularity from a line integral. – Wolfgang Bangerth Jun 6 '12 at 21:16
Thanks for your comments so far. To clarify: I typically have a set of points in the domain at which I know the function value as well as (analytically) its derivative and Laplacian values. I could obtain function values at new points in the domain (as well as the derivatives, etc., there), so could evaluate the Laplacian on a grid and search for its largest magnitude, but this would lack finesse and economy. I am wondering whether there is a robust estimate of the Laplacian's largest magnitude anywhere in the domain but using only the points already available. – jsl50 Jun 12 '12 at 10:04

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