This paper is devoted to the fast solution of interface concentrated finite element equations. The interface concentrated finite element schemes are constructed on the basis of a non-overlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain. Similar to data-sparse boundary element domain decomposition methods the total number of unknowns per subdomain behaves like $O((H/h)^{d−1})$, where H, h, and d denote the usual scaling parameter of the subdomains, the average discretization parameter of the subdomain boundaries, and the spatial dimension, respectively. We propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns. In particular, the so-called All-Floating Finite Element Tearing and Interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps.

Stuttgart, Univ., Diss., 2005.

The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, we derive convergence rates as known for the conforming finite element method in presence of regular solutions. Numerical examples illustrate the approach and the results.

The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, we derive convergence rates as known for the conforming finite element method in presence of regular solutions. Numerical examples illustrate the approach and the results.

This paper is devoted to the fast solution of interface concentrated finite element equations. The interface concentrated finite element schemes are constructed on the basis of a non-overlapping domain decomposition where a conforming boundary concentrated finite element approximation is used in every subdomain. Similar to data-sparse boundary element domain decomposition methods the total number of unknowns per subdomain behaves like $O((H/h)^{d−1})$, where H, h, and d denote the usual scaling parameter of the subdomains, the average discretization parameter of the subdomain boundaries, and the spatial dimension, respectively. We propose and analyze primal and dual substructuring iterative methods which asymptotically exhibit the same or at least almost the same complexity as the number of unknowns. In particular, the so-called All-Floating Finite Element Tearing and Interconnecting solvers are highly parallel and very robust with respect to large coefficient jumps.

Der GDSW-Vorkonditionierer ist ein zweistufiges überlappendes Schwarz-Gebietszerlegungsverfahren mit einem energieminimierenden Grobraum, dessen parallele Skalierbarkeit durch das direkt gelöste Grobproblem begrenzt ist. Zur Verbesserung der parallelen Skalierbarkeit wurde hier eine mehrstufige Erweiterung eingeführt. Für den Fall skalarer elliptischer Probleme wurde eine Konditionierungszahlschranke aufgestellt. Die parallele Implementierung wurde in das quelloffene ShyLU/FROSch Paket der Trilinos-Softwarebibliothek (http://trilinos.org) integriert und auf mehreren der leistungsstärksten Supercomputern der Welt (JUQUEEN, Forschungszentrum Jülich; SuperMUC-NG, LRZ Garching; Theta, Argonne Leadership Computing Facility, Argonne National Laboratory, USA) für Modellprobleme (Laplace und lineare Elastizität) getestet. Das angestrebte Ziel einer verbesserten parallelen Skalierbarkeit wurde erreicht, der Bereich der Skalierbarkeit wurde um mehr als eine Größenordnung erweitert. Die größten Rechnungen verwendeten mehr als 200000 Prozessorkerne des Theta Supercomputers. Zudem wurde die Anwendung des GDSW-Vorkonditionierers auf ein vollständig gekoppeltes nichtlineare Deformations-Diffusions Problem in der Chemomechanik betrachtet.