An iterative generalized Golub-Kahan algorithm for problems in structural mechanics - Arioli, Mario; Kruse, Carola; Rüde, Ulrich; Tardieu, Nicolas (arXiv, 2018)
This paper studies the Craig variant of the Golub-Kahan bidiagonalization algorithm as an iterative solver for linear systems with saddle point structure. Such symmetric indefinite systems in 2x2 block form arise in many applications, but standard iterative solvers are often found to perform poorly on them and robust preconditioners may not be available. Specifically, such systems arise in structural mechanics, when a semidefinite finite element stiffness matrix is augmented with linear multi-point constraints via Lagrange multipliers. Engineers often use such multi-point constraints to introduce boundary or coupling conditions into complex finite element models. The article will present a systematic convergence study of the Golub-Kahan algorithm for a sequence of test problems of increasing complexity, including concrete structures enforced with pretension cables and the coupled finite element model of a reactor containment building. When the systems are suitably transformed using augmented Lagrangians on the semidefinite block and when the constraint equations are properly scaled, the Golub-Kahan algorithm is found to exhibit excellent convergence that depends only weakly on the size of the model. The new algorithm is found to be robust in practical cases that are otherwise considered to be difficult for iterative solvers.<br

Implicitly extrapolated geometric multigrid on disk-like domains for the gyrokinetic Poisson equation from fusion plasma applications - Kühn, Martin J; Kruse, Carola; Rüde, Ulrich (SIAM Journal on Scientific Computing, 2021)
The gyrokinetic Poisson equation arises as a subproblem of Tokamak fusion reactor simulations. It is often posed on disk-like cross sections of the Tokamak that are represented in generalized polar coordinates. On the resulting curvilinear anisotropic meshes, we discretize the differential equation by finite differences or low order finite elements. Using an implicit extrapolation technique similar to multigrid tau-extrapolation, the approximation order can be increased. This technique can be naturally integrated in a matrix-free geometric multigrid algorithm. Special smoothers are developed to deal with the mesh anisotropy arising from the curvilinear coordinate system and mesh grading.<br