A Framework for Adaptive Multiscale Methods for Elliptic Problems

We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorithms for solving problems with no clear scale separation. We present numerical simulations demonstrating the effectiveness and adaptivity of the multiscale method, assess its computational complexity, and discuss the relationship between this framework and other multiscale methods, such as wavelets, multiscale finite element methods, and the use of harmonic coordinates. We prove in detail that the projection based method captures homogenization when there is strong scale separation.