# Project 2

### Multigrid Methods for Isogeometric Analysis

Isogeometric analysis (IGA), see [1], for discretizing (systems of) partial differential equations (PDEs) on a given computational domain provides an exact geometry representation of this domain as well as an accompanying finite dimensional subspace of functions on the computational domain for approximating the solution of the (system of) PDEs. An obvious advantage of this approach is that the geometry description of the computational domain, which is typically provided by a computer-aided design (CAD) system in terms of nonuniform rational B-splines (NURBS), can be directly used (without any further approximation process) for the discretization of the problem. Another important feature is based on the observation that the accompanying finite dimensional subspace can be rather lowdimensional for many interesting problems without losing the exact geometry representation of the computational domain. Refinement strategies (e.g.: h-, p-, k-refinement, see [1]) are available in order to refine the approximation space and, therefore, to improve the accuracy of the approximate solution. The approximation spaces generated by such refinement strategies are typically nested.

An efficient implementation of IGA for a (system of) PDE(s) requires an economic assembling process for setting up the system of discretized equations and a fast method for solving this system. In this project we will mainly focus on the second topic, the development and analysis of fast solvers. The IGA provides an ideal environment for geometric multigrid methods (GMG), a well-established class of solution techniques for the discretized (system of) PDEs, see [2]. So, the IGA brings a revival of GMG, which are better understood than their algebraic counterpart, the algebraic multigrid methods (AMG), which are more flexible in handling complex geometry but are less understood, see [3].

Possible applications are centered around optimal design of electric machines and of turbines. The underlying (systems of) PDEs of main interests are Maxwell's equations for magnetostatic problems and the incompressible Euler equations for rotational flow problems, as well as the optimality systems for associated optimization problems with Maxwell's equations or Euler equations as constraints (PDE-constrained optimization).

[1] T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg., 194(39-41):4135-4195, 2005. doi:doi:10.1016/j.cma.2004.10.008.

[2] W. Hackbusch. Multi-Grid Methods and Applications. Springer-Verlag, Berlin, 1985.

[3] U. Trottenberg, C. W. Oosterlee, and A. Schüller. Multigrid. With guest contributions by A. Brandt, P. Oswald, K. Stüben. Orlando, FL: Academic Press, 2001.