# Graph theory based preconditioners in isogeometric analysis

## MSc MTech Krishan Gahalaut

**Dec. 10, 2010, 2:30 p.m. HS 13**

Conjugate gradient method (CGM), and it's preconditioned version, are one of the most promising techniques for the solution of symmetric and positive definite linear system of equations. The number of iterations of the CGM depend on the ratio of largest to smallest eigenvalues. Support graph theory, introduced by Vaidya [1], is a methodology for bounding condition numbers of preconditioned systems. Specically, the extremal eigenvalues can be bounded with support graph techniques. Vaidya analyzed maximum weight spanning tree preconditioners, and Miller and Gremban [2], extended this work by using

support tree preconditioners.

Isogeometric analysis, introduced by Hughes et al. [3,4], is a novel approach to bridge the gap between geometry and numerical simulation. Broadly speaking, it replaces the polynomial based approximation in a finite element method by those functions, which are used to represent the geometry (e.g., NURBS, a well established methodology in computer aided design (CAD) community).

In this talk we discuss the methodology of graph theory based preconditioners for isogeometric analysis.