Coupling deal.II and FROSch: A Sustainable and Accessible (O)RAS Preconditioner

TL;DR

The overall software interface is straightforward and easy to use while giving satisfactory solver performances for challenging PDE systems.

Abstract

In this work, restricted additive Schwarz (RAS) and optimized restricted additive Schwarz (ORAS) preconditioners from the Trilinos package FROSch (Fast and Robust Overlapping Schwarz) are employed to solve model problems implemented using deal.II (differential equations analysis library). Therefore, a Tpetra-based interface for coupling deal.II and FROSch is implemented. While RAS preconditioners have been available before, ORAS preconditioners have been newly added to FROSch. The FROSch-deal.II interface works for both Lagrange-based and Nédélec finite elements. Here, as model problems, nonstationary, nonlinear, variational-monolithic fluid-structure interaction and the indefinite time-harmonic Maxwell's equations are considered. Several numerical experiments in two and three spatial dimensions confirm the performance of the preconditioners as well as the FROSch-deal.II interface. In conclusion, the overall software interface is straightforward and easy to use while giving satisfactory solver performances for challenging PDE systems.

Figures (14)

• Figure 2.1: Left: A square domain divided into two subdomains, where the size of both subdomains was increased by one layer. Right: A square domain divided into four subdomains, where the size of all subdomains was increased by one layer. In both cases, the overlap was computed based on the dual graph.
• Figure 5.1: Flow around an obstacle to which an elastic beam is attached. In the two-dimensional case, the domain is a rectangle, and the obstacle is modeled as a cycle with the center $C(0.20,0.20)$, and the lower right corner of the elastic beam is point $A(0.60,0.19)$. In the three-dimensional case, the domain is a rectangular cuboid with depth $D$. The obstacle is modeled as a cylinder with the center $C$, and the front lower right corner of the elastic beam is point $A$.
• Figure 5.2: FSI-1 results. Top left:$x$-displacement in $A(0.6,0.19)$. Top right:$y$-displacement in $A(0.6,0.19)$. Bottom left: Face drag measured around the cylinder and the elastic beam. Bottom right: Face lift measured around the cylinder and the elastic beam.
• Figure 5.3: The domain decomposition of the FSI-1 benchmark, as obtained by applying p4est. There are non-connected subdomains, as well as very long subdomains, which is less than optimal for a domain decomposition.
• Figure 5.4: Left: Plot of the resulting velocity into $x$-direction from the flow around the cylinder at time step $t=10\;\unit{s}$. Right: The mesh deformation at the tip of the elastic beam at time step $t=10\;\unit{s}.$

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4