A IETI-DP method for discontinuous Galerkin discretizations in Isogeometric Analysis with inexact local solvers

TL;DR

This paper develops an efficient IETI-DP solver for discontinuous Galerkin isogeometric discretizations, using inexact local solvers with tensor structures, leading to reduced memory and computational costs while maintaining robustness and scalability.

Contribution

It introduces an IETI-DP method with inexact local solvers for DG isogeometric discretizations, including a tensor-based approach and convergence analysis.

Findings

Condition number grows poly-logarithmically with grid size.

Convergence mildly depends on spline degree.

Significant memory reduction compared to direct solvers.

Abstract

We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines.