The fast adaptive composite grid (FAC) method for elliptic equation

The fast adaptive composite grid (FAG) 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