Goal-oriented space-time adaptivity for the Navier--Stokes equations based on the dual weighted residual method
This work provides a robust and efficient method for goal-oriented error control in fluid dynamics simulations, which is an incremental improvement over existing DWR-based approaches.
The authors developed a goal-oriented a posteriori error estimator using the Dual Weighted Residual method with space-time mesh adaptivity for the Navier-Stokes equations, achieving reliable control of target quantities on computationally feasible meshes. Benchmark computations demonstrated accuracy, efficiency, and stability.
This work presents a goal-oriented a posteriori error estimator based on the Dual Weighted Residual (DWR) method together with space-time mesh adaptivity for the Navier--Stokes equations. The resulting nonlinear algebraic systems on the space-time slabs are solved by Newton's method with GMRES, preconditioned by a slab-wise geometric multigrid method. This combination yields reliable control of target quantities on computationally feasible space-time meshes together with a robust and efficient solution of the algebraic systems. The implementation is based on a MPI-parallel programming model in the deal.II library. Further ingredients are a discontinuous Galerkin discretization in time and inf-sup stable finite element pairs with discontinuous pressure on tensor-product meshes. The performance of the approach is investigated in benchmark computations with regard to accuracy, efficiency, and stability.