In this paper we establish best approximation type error estimates for fully discrete Galerkin solutions of the time-dependent Stokes problem using the stream-function formulation. The resulting equations involve the spatial biharmonic operator and the time derivative of the Laplacian. 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 C¹ and C⁰ 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. A numerical example illustrates the theoretical findings.
Vexler et al. (Fri,) studied this question.