Fast Multipatch Isogeometric Analysis Solvers

MSc Christoph Hofer

July 13, 2018, 8:30 a.m. S2 416-1

This thesis is devoted to the generalization of the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method to linear algebraic systems arising from the Isogemetric Analysis (IgA) of linear elliptic boundary value problems, like stationary diffusion or heat conduction problems. This IgA version of the FETI-DP method is called Dual-Primal Isogeometric Tearing and Interconnect (IETI-DP) method. The FETI-DP method is well established as parallel solver for large-scale systems of finite element equations, especially, in the case of heterogeneous coefficients having jumps across subdomain interfaces. These methods belong to the class of non-overlapping domain decomposition methods.
In practise, a complicated domain can often not be represented by a single patch, instead a collection of patches is used to represent the computational domain, called multi-patch domains. Regarding the solver, it is a natural idea to use this already available decomposition into patches directly for the construction of a robust and parallel solver. We investigate the cases where the IgA spaces are continuous or even discontinuous across the patch interfaces, but smooth within the patches. In the latter case, a stable formulation is obtained by means of discontinuous Galerkin (dG) techniques. Such formulations are important for various reasons, e.g, if the IgA spaces are not matching across patch interfaces (different mesh-sizes, different spline degrees) or if the patches are not matching (gap and overlapping regions).
Using ideas from dG-FETI-DP methods, we extend IETI-DP methods in such a way that they can efficiently solve multi-patch dG-IgA schemes. This thesis also provides a theoretical foundation of IETI-DP methods. We prove the quasi-optimal dependence of the convergence behaviour on the mesh-size for both version. Moreover, the numerical experiments indicate robustness of these methods with respect to jumps in the coefficient and a weak dependence on the spline degree. All algorithms are implemented in the C++ library G+Smo.
Finally, this thesis investigates space-time methods for linear parabolic initial-boundary value problems, like instationary diffusion or heat conduction problems. The focus is again on efficient solution techniques. The aim is the development of solvers which are on the one hand robust with respect to certain parameters and on the other hand parallelizeable in space and time. We develop special block smoothers that lead to robust and efficient time-parallel multigrid solvers. The parallelization in space is again achieved by means of IETI-DP methods.