Optimal pressure approximation for the nonstationary Stokes problem by a variational method in time with post-processing

TL;DR

This paper develops a variational method with post-processing for the nonstationary Stokes problem, achieving optimal second order error estimates in pressure and velocity approximations in space and time.

Contribution

It introduces a novel pressure post-processing technique that ensures optimal error estimates for the nonstationary Stokes problem using higher order finite elements.

Findings

Optimal second order pressure error in time at midpoints

Optimal error estimates for velocity in space and time

Numerical tests confirm theoretical convergence rates

Abstract

We provide an error analysis for the solution of the nonstationary Stokes problem by a variational method in space and time. We use finite elements of higher order for the approximation in space and a Galerkin-Petrov method with first order polynomials for the approximation in time. We require global continuity of the discrete velocity trajectory in time, while allowing the discrete pressure trajectory to be discontinuous at the endpoints of the time intervals. We show existence and uniqueness of the discrete velocity solution, characterize the set of all discrete pressure solutions and prove an optimal second order estimate in time for the pressure error in the midpoints of the time intervals. The key result and innovation is the construction of approximations to the pressure trajectory by means of post-processing together with the proof of optimal order error estimates. We propose two variants for a post-processed pressure within the set of pressure solutions based on collocation techniques or interpolation. Both variants guarantee that the pressure error measured in the L2-norm converges with optimal second order in time and optimal order in space. For the discrete velocity solution, we prove error estimates of optimal order in time and space. We present some numerical tests to support our theoretical results.