This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix–vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evidence of its good behaviour with respect to the spline degree and the (possibly singular) geometry parametrization. We also extend the preconditioner to the multipatch case through an Additive Schwarz method.

本文在等几何分析的背景下，研究了与质量矩阵相关联的线性系统的快速解决方案。我们基于单变量参数质量矩阵的对角线比例Kronecker乘积，提出了既高效又易于实现的预处理器。它的应用比涉及质量矩阵本身的矩阵-矢量积更快。我们证明，随着网格尺寸的减小，预处理矩阵的条件数收敛到1，也就是说，预处理器渐近等效于精确逆。此外，我们给出了关于样条度和（可能是奇异的）几何参数化的良好行为的数值证据。我们还通过加性Schwarz方法将预处理器扩展到多修补程序的情况。