Enhanced domain decomposition Schwarz solution schemes for isogeometric collocation methods

Abstract

Isogeometric collocation methods have been introduced as an alternative to isogeometric Galerkin formulations, aiming at improving the computational cost of simulation by reducing the cost of assembly of the corresponding matrices. However, in contrast to their Galerkin counterparts, collocation formulations result in non-symmetric matrices of much higher dimensions, for a specified level of accuracy, which require special attention when addressing large-scale simulations. Furthermore, the presence of mixed boundary conditions may lead to indefinite collocation matrices, which hinder the convergence properties of domain decomposition-based iterative solution methods of the corresponding algebraic equations. To address these shortcomings, two-level hybrid, domain decomposition-based preconditioners are proposed, related to both the basic and the enhanced collocation formulations, in conjunction with the generalized minimal residual method and Schwarz-based preconditioners. The proposed solution schemes improve the computational efficiency of the isogeometric collocation simulations and exhibit numerical scalability for an increased number of subdomains. This is attributed to the fact that the iterations are performed on the Schur complement level of the corresponding matrices, leading to stable, computationally efficient, and robust solutions of the corresponding non-symmetric algebraic equations.