AbstractIn problems such as the computation of incompressible flows with moving boundaries, it may be necessary to solve Poisson's equation on a large sequence of related grids. In this paper the LU decomposition of the matrixA0representing Poisson's equation discretized on one grid is used to efficiently obtain an approximate solution on a perturbation of that grid. Instead of doing an LU decomposition of the new matrixA, the RHS is perturbed by a Taylor expansion ofA−1aboutA0. Each term in the resulting series requires one ‘backsolve’ using the originalLU.Tests using Laplace's equation on a square/rectangle deformation look promising; three and seven correction terms for deformations of 20% and 40% respectively yielded better than 1% accuracy.As another test, Poisson's equation was solved in an ellipse (fully developed flow in a duct) of aspect ratio 2/3 by perturbing about a circle; one correction term yielded better than 1% accuracy.Envisioned applications other than the computation of unsteady incompressible flow include: three‐dimensional parabolic problems in tubes of varying cross‐section, use of ‘elimination’ techniques other than LU decomposition, and the solution of PDEs other than Poiss