Locality and global information in elliptic solvers

We study how much global information a local approximation to an elliptic inverse needs. At each grid point, the solver receives nearby right-hand-side values and a continuous encoding of the full input; both the encoder and reconstruction may be nonlinear. For the Dirichlet Laplacian on an n^d grid, uniform relative energy error at any fixed tolerance below one requires order n^d/(w + 1)^d shared coordinates, where w is the local radius. A linear construction attains this order. We give finite bounds, leading constants as the tolerance approaches one, and a tradeoff between cumulative local propagation and the total dimension of global messages.

The lower bound combines a Borsuk-Ulam argument with canonical correlations of the Green matrix. For a prescribed linear measurement map, it also gives an explicit pair of inputs that the solver cannot distinguish. Experiments on the two-dimensional Poisson problem compare local convolutional, moment-based, Fourier, and multiscale models. A model using 16 sine moments has mean energy error 6.1% on smooth forcings and 67.0% on white forcings at the training resolution. Broader training reduces some of this gap, whereas none of the tested learned initial guesses reduces median end-to-end solve time at the largest timed grid.