Kronecker Products on Preconditioning

Abstract

Numerical techniques for linear systems arising from discretization of partial differential equations are nowadays essential for understanding the physical world. Among these techniques, iterative methods and the accompanying preconditioning techniques have become increasingly popular due to their great potential on large scale computation.

In this work, we present preconditioning techniques for linear systems built with tensor product basis functions. Efficient algorithms are designed for various problems by exploiting the Kronecker product structure in the matrices, inherited from tensor product basis functions.

Specifically, we design preconditioners for mass matrices to remove the complexity from the basis functions used in isogeometric analysis, obtaining numerical performance independent of mesh size, polynomial order and continuity order; we also present a compound iteration preconditioner for stiffness matrices in two dimensions, obtaining fast convergence speed; lastly, for the Helmholtz problem, we present a strategy to `hide' its indefiniteness from Krylov subspace methods by eliminating the part of initial error that corresponds to those negative generalized eigenvalues. For all three cases, the Kronecker product structure in the matrices is exploited to achieve high computational efficiency.