Adaptive variational multiscale methods
In this thesis we present a new adaptive multiscale method for solving elliptic partial differential equations. The method is based on numerical solution of decoupled local fine scale problems on patches. Critical parameters such as fine and coarse scale mesh size and patch size are tuned automatically by an adaptive algorithm
based on a posteriori error estimates. We extend the method to a mixed formulation of the Poisson equation and derive error estimates in this case as well.
We also present a framework for adaptivity based on a posteriori error estimates for multi-physics problems. We study a coupled flow and transport problem and derive an a posteriori error estimate for a linear functional by introducing two dual problems, one associated
with the transport equation and one associated with the flow
equation. We also apply this method to a model problem in oil reservoir simulation.
finite element method
a posteriori error estimation
variational multiscale method
oil reservoir simulation