$L^2(I;H^1(Ω))$ and $L^2(I;L^2(Ω))$ best approximation type error estimates for Galerkin solutions of transient Stokes problems

Abstract: In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in $L^2(I;L^2(\Omega)^d)$ and $L^2(I;H^1(\Omega)^d)$ norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type $L^2(I;H^1(\Omega))$ error estimates seems to be new even for scalar parabolic problems.