# Isogeometric Methods for Elliptic Partial Differential Equations: Condition number Estimates; Fast iterative solvers

## MSc MTech Krishan Gahalaut

**Oct. 30, 2012, 4 p.m. S2 059**

We shall derive the bounds for the extremal eigenvalues and the spectral condition

number of matrices for isogeometric discretizations of elliptic partial differential

equations in Ω ε R^d, d = 2, 3. For the h-refinement, the condition number of the

stiffness matrix is bounded above and below by a constant times h^-2, and the condition

number of the mass matrix is uniformly bounded. For the p-refinement, the condition

number is bounded above by p^2d4^pd and p^2(d-1) 4^pd for the stiffness matrix and the

mass matrix respectively. Numerical results supporting the theoretical estimates will

be presented. Some numerical results on the condition number for varying smoothness

of the basis functions will also be discussed.

We recently introduced geometric multigrid methods for isogeometric discretizations

[2], as a next step we shall discuss algebraic multilevel iteration (AMLI) method

for solving linear system arising from the isogeometric discretization. Theoretical

bounds for the constant

in the strengthened Cauchy-Bunyakowski-Schwarz inequality

will be discussed. For a fixed p, the constant

will be analyzed for different regularities

of the B-spline basis functions.