Goal-oriented space-time adaptivity and Efficient Data Structures for convection-dominated problems

M.P. Bruchhäuser and U. Köcher

Jan. 14, 2020, 3:30 p.m. S2 054

The numerical approximation of instationary convection-diffusion-reaction problems
\begin{equation}
\label{modell}
\partial_t u - \nabla \cdot (\varepsilon \nabla u) + \boldsymbol{b} \cdot \nabla u
+ \alpha u
= f
\end{equation}
with small diffusion $0<\varepsilon \ll 1$ remains to be a challenging task.
Solving those problems numerically naively by standard Galerkin finite element
approximations leads to perturbed solutions with spurious unphysical oscillations
close to regions with sharp inner or boundary layers.
To reduce oscillations stabilization concepts are applied to the finite element
approximations. Besides stabilization concepts, the application of adaptive mesh
refinement has been considered in the past to further eliminate oscillations.

In this contribution we present a space-time adaptive algorithm based on the
Dual Weighted Residual method as well as the underlying software design for
the implementation process.
Flexible data structures for the open source deal.II library
are presented that are indispensable to handle the complex framework for solving
the primal and dual problem.
In numerical experiments its applicability is studied and demonstrated for
benchmark problems of convection-dominated transport;
cf. \cite{BruchhaeuserBBS18,BruchhaeuserKBB18}.

\begin{thebibliography}{0}
\itemsep0ex
\small
\bibitem{BruchhaeuserBBS18a}
Bruchh\"auser, M.P., Schwegler, K., Bause, M.: Dual weighted residual based
error control for nonstationary convection-dominated equations: potential or
ballast?, LNCSE, accepted (2019), pp. 1-13. arXiv:1812.06810.

\bibitem{BruchhaeuserKBB18}
K\"ocher, U., Bruchh\"auser, M.P., Bause, M.: Efficient and scalable data
structures and algorithms for goal-oriented adaptivity of space-time
FEM codes, SoftwareX, 10 (2019), 100239, https://doi.org/10.1016/j.softx.2019.100239

\bibitem{BruchhaeuserBBS18}
Bruchh\"auser, M.P., Schwegler, K., Bause, M.: Numerical study of goal-oriented
error control for stabilized finite element methods. In Apel, T. et al. (eds.)
Advanced Finite Element Methods with Applications. FEM 2017, Lecture Notes in
Computational Science and Engineering, vol. 128, Springer, Cham, pp. 85-106
(2019), https://doi:10.1007/978-3-030-14244-5\_5
\end{thebibliography}