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

## MSc MTech Krishan Gahalaut

**Oct. 30, 2012, 3 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 $Ω \in R^d, d = 2, 3$. 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.