Marius Paul Bruchhäuser

Marius Paul Bruchhäuser, Kristina Schwegler, Markus Bause

Even though substantial progress has been made in the numerical approximation of convection-dominated problems, its major challenges remain in the scope of current research. In particular, parameter robust a posteriori error estimates for quantities of physical interest and adaptive mesh refinement strategies with proved convergence are still missing. Here, we study numerically the potential of the Dual Weighted Residual (DWR) approach applied to stabilized finite element methods to further enhance the quality of approximations. The impact of a strict application of the DWR methodology is particularly focused rather than the reduction of computational costs for solving the dual problem by interpolation or localization.

Kristina Schwegler, Marius P. Bruchhäuser, Markus Bause

The numerical approximation of convection-dominated problems continues to remain subject of strong interest. Families of stabilization techniques for finite element methods were developed in the past. Adaptive techniques based on a posteriori error estimates offer potential for further improvements. However, there is still a lack in robust a posteriori error estimates in natural norms of the discretizations. Here we combine the dual weighted residual method for goal-oriented error control with stabilized finite element approximations. By a duality argument an error representation is derived on that a space-time adaptive approach is built. It differs from former works on the dual weighted residual method. Numerical experiments illustrate that our schemes are capable to resolve layers and sharp fronts with high accuracy and to further reduce spurious oscillations of approximations.

Marius Paul Bruchhäuser, Nils Margenberg, Markus Bause

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.