MATHEMATICS OF COMPUTATION VOLUME 46, NUMBER 174 APRIL 1986, PAGES 439-456 The Fast Adaptive Composite Grid (FAC) Method for Elliptic Equations* By S. McCormick and J. Thomas Abstract. The fast adaptive composite grid (FAC) method is a systematic process for solving differential boundary value problems. FAC uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution. It can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation. This paper develops both the theoretical and practical aspects of FAC as it applies to elliptic problems. 1. Introduction. The need for local resolution in physical models occurs frequently in practice. Special local features of the forcing function, operator coefficients, boundary, and boundary conditions can demand resolution in restricted regions of the domain that is much finer than the required global resolution. It is important that the discretization and solution processes account for this locally, that is, that the local phenomena do not precipitate a dramatic increase in the overall computation. Unfortunately, this objective of efficiently adapting to local features is often in conflict with the solution process: equation solvers can degrade or even fail to apply in the presence of varying discretization scales; data structures that account for irregular grids can be cumbersome; the computer architecture may not be able to effectively account for grid irregularity (e.g., " vectorizability" may be inhibited); etc. In fact, even the discretization process itself may find difficulty with this objective: for finite differences, it is problematic to develop accurate difference formulae for irregular grids; for finite elements, this objective is reflected in the substantial overhead costs needed to automate the discretization. The fast adaptive composite grid method (FAC [11]) is a discretization and solution method designed to achieve efficient local resolution by constructing the discretization based on various regular grids and using these grids as a basis for fast solution. Its basic computational objective is to solve a "good" discretization on an irregular grid by way of regular grids only. Its basic assumption is that both discretization and solution on regular grids are easy by comparison. Although FAC is in essence very similar to multi-level adaptive techniques (MLAT; cf. [1], [3], [5], [10])and local defect correction (LDC; cf. [8]),it differs in several simple but important respects. First, FAC is a systematic approach that Received July 10, 1984; revised August 2, 1985. 1980 Mathematics Subject Classification.Primary 65N20; Secondary 65N30. This work was supported by AFOSR grant number FQ8671-83-01322 and the National Bureau of Standards. ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page 439 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use