Isogeometric Analysis: Condition Number Estimates and Fast Solvers

Isogeometric methods were introduced by Hughes et al. in 2005. Since its introduction these methods have been used in many practical problems in science and engineering. However, so far, only few papers appeared in the field of iterative solvers for these methods. This thesis contributes to the development of fast iterative solvers for the matrices arising in isogeometric discretization of elliptic partial differential equations. Since the performance of iterative solvers depends on the properties of the coefficient matrix, first we focus on the study of condition number estimates for the isogeometric matrices. The bounds for the extremal eigenvalues and the spectral condition number of matrices arising in isogeometric discretizations of elliptic partial differential equations in Ω ⊂ R, d = 2, 3, are given in this thesis. Using the results from Bazilevs et al. on approximation properties, stability analysis and error estimates for isogeometric discretizations, and existing finite element theory, we obtain the condition number estimates for h-refined meshes. 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, it is proved that the condition number is bounded above by p4 and p2(d−1)4pd for the stiffness matrix and the mass matrix, respectively. For large problem size, the high condition number of coefficient matrix necessitates the development of fast and robust iterative solvers. In this thesis, we present three classes of iterative solvers. At first, the multigrid methods for isogeometric discretization are presented. The smoothing property of the relaxation method, and the approximation property of the intergrid transfer operators are analyzed for two-grid and multi-grid cycles. It is shown that the convergence of the multigrid solver is independent of the discretization parameter h, and that the overall solver is of optimal complexity. Secondly, we present another class of iterative solvers called algebraic multilevel iteration (AMLI) methods for isogeometric discretizations. The formation of coarse space and complement hierarchical space is discussed for B-splines and NURBS. The numerical study of Cauchy-Bunyakowski-Schwarz constant γ, measuring the quality of splitting between coarse space and its hierarchical complement, is also presented. It is found that the convergence of the AMLI solver is independent of the discretization parameter h, and for Cp−1 continuous basis functions the convergence rates are independent of polynomial degree p. For C continuous basis functions, numerical results indicates almost p-independent convergence rates. Thirdly, some numerical results on graph theory based preconditioners, namely, Vaidya’s preconditioners (maximum weight spanning tree) and Gremban and Miller’s preconditioners (support tree) are also presented. However, these preconditioners do not yield h-independent results for the condition number of preconditioned system.