Approximations of negative Sobolev norms

Consider the standard Cahn-Hilliard free energy, augmented by a nonlocal interaction term which measures the $H^{-1}$ norm of a zero-mean function. Could someone point me to a reference where this nonlocal term is numerically approximated for a function on a compact domain, but without assuming periodicity? The periodic case is handled, for example, in a paper by Choksi et al in SIAM J. Appl.Math., 2009. Specifically, any strategies which avoid a Poisson solve would be welcome.

flag	asked Apr 29 2011 at 3:59 Nilima Nigam 976●6●14

Since posting this question, I've tried the following two approaches: (a) using a quadrature to compute the mean of $u$ (denoted $A$),solved the Poisson problem $−\Delta v=u−A$, and used $\|\nabla v\|_0$ as an approximation for $\|u-A\|_{-1}$. This entailed a Poisson solve. (b) place a large grid around the compact support of $u$, and use a quadrature to estimate the desired norm using the Slobodeckii definition. Both methods appear inefficient. I have $O(N^2)$ versions where $N=\#$ of nodes, in the absence of FFTs. There must be a better way to do this. – Nilima Nigam Jun 4 2011 at 1:31

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