Kristina Schwegler

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.

Marius Paul Bruchhäuser, Kristina Schwegler, Markus Bause

The efficient and reliable approximation of convection-dominated problems continues to remain a challenging task. To overcome the difficulties associated with the discretization of convection-dominated equations, stabilization techniques and a posteriori error control mechanisms with mesh adaptivity were developed and studied in the past. Nevertheless, the derivation of robust a posteriori error estimates for standard quantities and in computable norms is still an unresolved problem and under investigation. Here we combine the Dual Weighted Residual (DWR) method for goal-oriented error control with stabilized finite element methods. By a duality argument an error representation is derived on that an adaptive strategy is built. The key ingredient of this work is the application of a higher order discretization of the dual problem in order to make a robust error control for user-chosen quantities of interest feasible. By numerical experiments in 2D and 3D we illustrate that this interpretation of the DWR methodology is capable to resolve layers and sharp fronts with high accuracy and to further reduce spurious oscillations.

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.