# Algebraic multilevel iteration method for isogeometric discretization

## MSc MTech Krishan Gahalaut

**June 5, 2013, 1:30 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 hierarchical splitting of the solution space. We present the multilevel structure of B-splines and NURBS spaces and their corresponding hierarchical spaces. The matrix formulation of coarse grid operators and its hierarchical 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 strengthened 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.