Our ability to numerically simulate physical processes is severely constrained by our ability to solve the complex linear systems that are often at the core of the computation. Multigrid methods offer an efficient solution technique for many such problems. However, fixed multigrid processes are based on an overall assumption of smoothness that may not be appropriate for a given problem. Our aim is to develop an adaptive multigrid scheme that replaces this predetermined sense of smoothness by one that is determined automatically. This paper focuses on the principal component of such a scheme: adaptive interpolation. Our method is based on computing a representative error component that is not quickly reduced by relaxation and fitting interpolation so that it is eliminated by the coarse-grid correction process. Numerical results are given to support the efficiency of this approach.
This research has been performed in collaboration with Marian Brezina, Tom Manteuffel, Steve McCormick, and John Ruge at CU-Boulder, and Rob Falgout at CASC-LLNL. It has been supported by an NSF SciDAC grant (TOPS).