A Posteriori Error Estimation for the Fractional Step Theta Discretization of the Incompressible Navier-Stokes Equations

In this work, we derive a goal-oriented a posteriori error estimator for the error due to time discretization. As time discretization scheme we consider the fractional step theta method, that consists of three subsequent steps of the one-step theta method. In every sub-step, the full incompressible system has to be solved (in contrast to time integrators of operator splitting type). The resulting fractional step theta method combines various desirable properties like second order accuracy, strong A-stability and very little numerical dissipation. The derived error estimator is based on a mathematical trick: we define an intermediate time-discretization scheme based on a Petrov-Galerkin formulation. This method is up to a numerical quadrature error equivalent to the theta time stepping scheme. The error estimator is assembled as one weighted residual term given by the Dual Weighted Residual method measuring the error between real solution and solution to the Petrov-Galerkin formulation (that at no time has to be calculated) and one additional residual estimating the discrepancy between actual time stepping scheme used for simulation and the intermediate Petrov-Galerkin formulation.