The cost of continuity: Performance of iterative solvers on isogeometric finite elements

Abstract

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using Co B-splines, which span traditional finite element spaces, and Cp-1 B-splines, which represent maximum continuity We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most 33p2/8 times more expensive for the more continuous space, although for moderately low p, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high p. Preconditioning options can be up to p3 times more expensive to set up, although this difference significantly decreases for some popular preconditioners such as incomplete LU factorization. © 2013 Society for Industrial and Applied Mathematics.