Finite element methods for the Stokes problem on complicated domains

Abstract

It is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process. In case of complicated domains appearing in many applications such as ground water flows this requirement sometimes becomes a bottleneck. The resolution condition links the computational complexity a priorily to the number (and size) of geometric details. Therefore even the coarsest available discretization can lead to a huge number of unknowns. In this paper, we will relax the resolution condition and introduce coarse (optimal order) approximation spaces for Stokes problems on complex domains. The described method will be efficient in the sense that the number of unknowns is only linked to the properties of the solution and not to the problem data. The presentation picks up the concept of composite finite elements for the Stokes problem presented in a previous paper of the authors. Here, the a priori error and stability analysis of the proposed mixed method is generalized to quite general, i.e. slip and leak boundary conditions that are of great importance in practical applications.