Matrix-Free Inexact Preconditioning Techniques for Isogeometric Tensor-Product Discretizations

We propose a matrix-free inexact preconditioning strategy for elliptic partial differential equations discretized by the isogeometric Galerkin method on tensor-product spline spaces. We base our preconditioner on an approximation of the discrete linear operator by a sum of Kronecker product matrices. The action of its inverse on a vector of coefficients is approximated by an inner preconditioned conjugate gradient solve. The forward problem is solved by the inexact preconditioned conjugate gradient method. The complexity of the Kronecker matrix-vector products in the inner iteration is lower than the complexity of the matrix-vector products in the forward problem, leading to a reduced number of iterations and significant performance gains. We show the robustness, efficiency and effectiveness of our approach in test problems involving the Poisson equation and linear elasticity, and illustrate the performance gain with respect to preconditioning techniques based on fast diagonalization. The proposed method is implemented in our open-source Julia framework for spline based discretization methods.