Computation of volume potentials on structured grids with the method of local corrections

We present a new version of the Method of Local Corrections (MLC) \cite{mlc}, a multilevel, low communications, non-iterative, domain decomposition algorithm for the numerical solution of the free space Poisson's equation in 3D on locally-structured grids. In this method, the field is computed as a linear superposition of local fields induced by charges on rectangular patches of size $O(1)$ mesh points, with the global coupling represented by a coarse grid solution using a right-hand side computed from the local solutions. In the present method, the local convolutions are further decomposed into a short-range contribution computed by convolution with the discrete Green's function for an $Q^{th}$-order accurate finite difference approximation to the Laplacian with the full right-hand side on the patch, combined with a longer-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 method with 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 radius of the support of the local solutions scaled by $h$. Thus we have eliminated the low-order accuracy of the original method (which corresponds to $P=1$ in the present method) for smooth solutions, while keeping the computational cost per patch nearly the same with that of the original method. Specifically, in addition to the local solves of the original method we only have to compute and communicate the expansion coefficients of local expansions (that is, for instance, 20 scalars per patch for $P=4$). Several numerical examples are presented to illustrate the new method and demonstrate its convergence properties.