Speaker
Description
We address the optimal order approximation of the pressure trajectory for an equal-order in time variational discretization of velocity and pressure in Stokes systems. In the literature, the pressure approximation of Stokes and Navier-Stokes systems has attracted less attention than the velocity approximation, even though being of equal importance for applications, for instance, for the computation of the drag and lift coefficient of flows around obstacles. The difficulties in the pressure approximation arise from the saddle point structure of the Stokes system and the lack of information regarding the computation of discrete initial pressure values [J. Heywood, R. Rannacher: Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), pp. 275–311.]
We consider approximations of the Stokes equations by space-time finite element methods (STFEMs) of equal order in time. For brevity, piecewise linear approaches are studied here. STFEMs feature the natural construction of higher order discretization schemes for partial differential equations and coupled systems [M. Anselmann, M. Bause, A geometric multigrid method for space-time finite element discretizations of the Navier–Stokes equations and its application to 3d flow simulation, ACM Trans. Math. Softw., 49 (2023), Article No.: 5, pp. 1–25; https://doi.org/10.1145/3582492]. We show that the pressure trajectory is not defined uniquely. We propose two variants for a post-processed pressure in the set of the pressure solutions which guarantee for the pressure error in the L2-norm optimal second order estimates in time and optimal order estimates in space. The first-one yields a globally continuous trajectory and is based on collocation conditions that are imposed in the discrete time nodes. The second one uses the idea of interpolation based on the accurate discrete pressure values in the Gauss nodes (midpoints of the subintervals). This approach leads to a global pressure trajectory that is in general discontinuous at the endpoints of the time intervals. The post-processing and error analysis are presented [M. Anselmann, M. Bause, G. Matthies, F. Schieweck, Optimal order pressure approximation for the Stokes problem by a variational method in time with post-processing, in progress (2024), pp. 1-28]. The theoretical findings are illustrated by numerical computations.
