BDDC Deluxe Algorithms for Two-Dimensional H($curl$) Isogeometric Analysis

Abstract

Isogeometric analysis has been introduced as an alternative to finite elements to simplify the integration of computer-aided design and the discretization of variational problems of continuum mechanics. In contrast to finite elements, the basis functions of isogeometric analysis, B-splines and nonuniform rational basis splines, are typically not nodal which makes the interface between subdomains fat with several layers of knots and which creates new issues in the design and analysis of iterative solvers based on domain decomposition methods. The resulting systems of algebraic equations also tend to be much more ill-conditioned than those derived from finite elements. The deluxe variant of adaptive balancing domain decomposition by constraints algorithms have proven successful, and in this study, it is applied to two-dimensional problems formulated in H($curl$). Numerical results show very good performance even for high degrees of the underlying B-splines, and fast convergence has been found for $p \leq 7$ using no more than $2(p^2+p-1)$ global (primal) variables per subdomain, where $p$ is the degree of the B-splines. All our results are focused on the most difficult case with the maximum degree of continuity of the B-splines.