$L^2(I;H^1(\mathrm{\Omega }))$ and $L^2(I;L^2(\mathrm{\Omega }))$ best approximation type error estimates for Galerkin solutions of transient Stokes problems

In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in $L^2(I;L^2(\mathrm{\Omega }{)}^d)$ and $L^2(I;H^1(\mathrm{\Omega }{)}^d)$ 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 $L^2(I;H^1(\mathrm{\Omega }))$ error estimates seems to be new even for scalar parabolic problems.