  A number of different multiscale methods have been developed as a robust
alternative to upscaling and as a means for accelerated reservoir simulation of
high-resolution geomodels. In their basic setup, multiscale methods use a
restriction operator to construct a reduced system of flow equations on a
coarser grid, and a prolongation operator to map pressure unknowns from the
coarse grid back to the original simulation grid. The prolongation operator
consists of basis functions computed numerically by solving localized flow
problems. One can use the resulting multiscale solver both as a
CPR-preconditioner in fully implicit simulators or as an efficient approximate
iterative linear solver in a sequential setting. The latter approach has been
successful implemented in a commercial simulator. Recently, we have shown that
you can obtain significantly faster convergence if you instead of using a
single pair of prolongation-restriction operators apply a sequence of such
operators, where some of the operators adapt to faults, fractures, facies, or
other geobodies. Herein, we present how you can accelerate the convergence even
further, if you also include additional basis functions that capture local
changes in the pressure.
