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.