Adaptive Variational Multiscale Methods Based on A Posteriori Error Estimation: Duality Techniques for Elliptic Problems

The variational multiscale method (VMM) provides a general framework for construction of multiscale finite element methods. In this paper we propose a method for parallel solution of the fine scale problem based on localized Dirichlet problems which are solved numerically. Next we present a posteriori error representation formulas for VMM which relates the error in linear functionals to the discretization errors, resolution and size of patches in the localized problems, in the fine scale approximation. These formulas are derived by using duality techniques. Based on the a posteriori error representation formula we propose an adaptive VMM with automatic tuning of the critical parameters. We primarily study elliptic second order partial differential equations with highly oscillating coefficients or localized singularities.