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We present a new version of the Method of Local Corrections (MLC) \cite{mlc},a multilevel, low communications, non-iterative, domain decomposition algorithmfor the numerical solution of the free space Poisson's equation in 3D onlocally-structured grids. In this method, the field is computed as a linearsuperposition of local fields induced by charges on rectangular patches of size$O(1)$ mesh points, with the global coupling represented by a coarse gridsolution using a right-hand side computed from the local solutions. In thepresent method, the local convolutions are further decomposed into ashort-range contribution computed by convolution with the discrete Green'sfunction for an $Q^{th}$-order accurate finite difference approximation to theLaplacian with the full right-hand side on the patch, combined with alonger-range component that is the field induced by the terms up to order $P-1$of the Legendre expansion of the charge over the patch. This leads to a methodwith a solution error that has an asymptotic bound of $O(h^P) + O(h^Q) +O(\epsilon h^2) + O(\epsilon)$, where $h$ is the mesh spacing, and $\epsilon$is the max norm of the charge times a rapidly-decaying function of the radiusof the support of the local solutions scaled by $h$. Thus we have eliminatedthe low-order accuracy of the original method (which corresponds to $P=1$ inthe present method) for smooth solutions, while keeping the computational costper patch nearly the same with that of the original method. Specifically, inaddition to the local solves of the original method we only have to compute andcommunicate the expansion coefficients of local expansions (that is, forinstance, 20 scalars per patch for $P=4$). Several numerical examples arepresented to illustrate the new method and demonstrate its convergenceproperties.

Computation of Volume Potentials on Structured Grids Using the Method of Local Corrections