Adaptive Galerkin finite element methods for partial differential equations

We present a general method for error control and mesh adaptivity in Galerkin finite element discretizations of partial differential equations. Our approach is based on the variational framework of projection methods and uses concepts from optimal control and model reduction. By employing global duality arguments and Galerkin orthogonality, we derive a posteriori error estimates for quantities of physical interest. These residual-based estimates contain the dual solution and provide the basis of a feed-back process for successive mesh adaptation. This approach is developed within an abstract setting and illustrated by examples for its application to different types of differential equations including also an optimal control problem.