Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity with Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems
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In this presentation, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier-Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor-Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition of unity based error localization. We derive the space-time Newton method for the Navier-Stokes equations and substantiate our methodology on 2D benchmark problems from computational fluid mechanics: flow around a cylinder and flow over backward facing step. The presentation concludes with current developments and some preliminary findings of extensions to our space-time adaptivity framework.
