Fully discrete error analysis of finite element discretizations of time-dependent Stokes equations in a stream-function formulation

In this paper we establish best approximation type error estimates for the fully discrete Galerkin solutions of the time-dependent Stokes problem using the stream-function formulation. For the time discretization we use the discontinuous Galerkin method of arbitrary degree, whereas we present the space discretization in a general framework. This makes our result applicable for a wide variety of space discretization methods, provided some Galerkin orthogonality conditions are satisfied. As an example, conformal $C^1$ and $C^0$ interior penalty methods are covered by our analysis. The results do not require any additional regularity assumptions beyond the natural regularity given by the domain and data and can be used for optimal control problems.