Easy and Efficient preconditioning of the Isogeometric Mass Matrix

TL;DR

This paper introduces a simple, efficient preconditioner for the isogeometric mass matrix that accelerates linear system solutions, with proven convergence properties and good performance across various geometries and spline degrees.

Contribution

It proposes a diagonal-scaled Kronecker product preconditioner that is easy to implement, faster than direct matrix-vector products, and asymptotically equivalent to the inverse of the mass matrix.

Findings

Condition number converges to 1 as mesh size decreases

Preconditioner performs well across different spline degrees

Effective extension to multipatch geometries

Abstract

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.