Fully discrete best approximation type estimates in $L^{\infty}(I;L^2(\Omega)^d)$ for finite element discretizations of the transient Stokes equations

TL;DR

This paper establishes optimal error estimates for fully discrete finite element methods solving the transient Stokes equations, using discontinuous Galerkin in time and standard finite elements in space, without requiring extra solution smoothness.

Contribution

It provides the first optimal best approximation error estimates for fully discrete schemes for the transient Stokes problem under minimal assumptions.

Findings

Optimal error bounds in $L^{ abla}(I;L^2( abla)^d)$ for finite element discretizations.

Analysis based on discrete maximal parabolic regularity.

No additional smoothness assumptions on the data or solutions.

Abstract

In this article we obtain an optimal best approximation type result for fully discrete approximations of the transient Stokes problem. For the time discretization we use the discontinuous Galerkin method and for the spatial discretization we use standard finite elements for the Stokes problem satisfying the discrete inf-sup condition. The analysis uses the technique of discrete maximal parabolic regularity. The results require only natural assumptions on the data and do not assume any additional smoothness of the solutions.