Efficient Solution Strategies in Isogeometric Analysis

Numerical computation of solutions for partial differential equations plays an important role in modern product development and engineering. Also, the use of computer aided design (CAD) software in the design process has become a widespread standard. These two fields are closely connected in the practical development process, but the corresponding techniques have been developed independently over the last decades, and a gap has opened between them. Transferring information between these two fields and processing transferred data to fit the respective requirements can be a very costly procedure in practical applications. Isogeometric analysis (IGA) aims at closing this gap. By directly using the geometry representation from CAD, the need of transforming geometry data is eliminated. By using underlying non-uniform rational B-splines (NURBS) as ansatz functions for numerical solutions, an initial mesh is obtained automatically, thereby eliminating the need of creating a new mesh of the imported object. Furthermore, one can profit from certain properties of NURBS functions, such as high regularity and NURBS-specific refinement options. In this thesis, two particular aspects, which arise in the course of numerical computations, are addressed in the context of IGA. In the first part, we consider the situation where a complicated object cannot be represented by a single NURBS geometry mapping, and is thus composed of several subdomains. By applying techniques from finite element tearing and interconnecting methods in isogeometric analysis, we introduce the isogeometric tearing and interconnecting (IETI) method. We discuss requirements for and the realization of C-coupling at subdomain interfaces, as well as suitable preconditioners, both for fully-matching settings and for situations with socalled “hanging knots”. The latter appear in local refinement methods which are introduced by the IETI method, and which are also discussed in this thesis. In the second part, we address the issue of quantitative a posteriori error estimation. The presently used error estimation techniques are adapted from classical finite element methods and do not take advantage of IGA-specific properties. Furthermore, these estimators do not provide sharp quantitative error bounds. By applying functional-type a posteriori error estimators in IGA, we