Graph theory based preconditioners in isogeometric analysis

MSc MTech Krishan Gahalaut

Dec. 10, 2010, 3:30 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 posi-
tive 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 ap-
proach 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) com-
munity).
In this talk we discuss the methodology of graph theory based precondition-
ers for isogeometric analysis.