A Mixed Adaptive Variational Multiscale Method

We present a mixed adaptive variational multiscale method for solving elliptic second order problems. This work is an extension of the adaptive variational multiscale method (AVMS), introduced by Larson and Malqvist, to a mixed formulation. The method is based on a particular splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part based on solution of decoupled localized subgrid problems. We present the mixed AVMS method and derive a posteriori error estimates for both linear functionals and the energy norm. Based on the estimates we propose an adaptive algorithm for automatic tuning of critical discretization parameters. Finally, we present numerical examples on a two dimensional slice of an oil reservoir.