# Algebraic multilevel iteration method for isogeometric discretization

## MSc MTech Krishan Gahalaut

**June 5, 2013, 3:30 p.m. S2 059**

The isogeometric analysis proposed by Hughes et al. in [1], has received

great deal of attention in the computational mechanics community. In this

talk we shall present algebraic multilevel iteration (AMLI) methods [2, 3]

for solving linear system arising from the isogeometric discretization of

elliptic boundary value problems. AMLI methods are based upon the hi-

erarchical splitting of the solution space. We present the multilevel struc-

ture of B-splines and NURBS spaces and their corresponding hierarchical

spaces. The matrix formulation of coarse grid operators and its hierar-

chical complementary operators will be discussed for varying regularity of

B-spline basis functions. For NURBS, we generate these operators from

B-splines and the corresponding weights. We shall discuss the quality of

splitting of spaces which is measured by the constant $\gamma$ in the strength-

ened Cauchy-Bunyakowski-Schwarz (CBS) inequality. For a fixed p, the

constant $\gamma$ will be analyzed for different regularities of the B-spline basis

functions. AMLI methods when applied in the framework of isogeometric

analysis shows h-independent convergence rates. Supporting numerical

results for CBS constant $\gamma$, and convergence factor and iterations count

for linear AMLI V -cycles and W-cycle, and for nonlinear AMLI W-cycle

are provided. Numerical results also show that these methods exhibit

almost p-independent convergence rates. Numerical tests are performed,

in two-dimensions on square domain and quarter annulus, and in three-

dimensions on thick ring. Moreover, for a uniform mesh on a unit interval,

the explicit representation of B-spline basis functions for a fixed mesh size

h is given for p = 2; 3; 4 and for $C^0$ and $C^{p-1}$ smoothness.