Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems

We develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution. The fine scale is approximated by a sum of solutions to decoupled localized problems, which are solved numerically on a fine grid partition of a patch of coarse grid elements. The sizes of the patches of elements may be increased to control the error caused by localization. We derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters: the coarse grid mesh size, the fine grid mesh size, and the sizes of the patches. Based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters. Finally, we show how the method works in practice by presenting various numerical examples.