Dual-primal Isogeometric Tearing and Interconnecting Solvers for adaptively refined multi-patch configurations
Stefan Takacs, Stefan Tyoler
TL;DR
This work extends the dual-primal Isogeometric Tearing and Interconnecting (IETI-DP) framework to adaptively refined, non-matching multi-patch IgA geometries that produce T-junctions, ensuring solvability through primal constraints and a skeleton-based Schur complement. A scalable preconditioner with selection scaling is analyzed, yielding a condition-number bound of κ(M_{sD}F) ≤ C p max{ max_{ℓ≺k} ν^{(k)}/ν^{(ℓ)}, 1 } (1+ log p + max_k log(H_k/h_k))^2, under admissibility and Schönberg–Whitney-type assumptions; diffusion jumps can affect robustness, motivating remedies such as deluxe preconditioning or adaptive refinement. Numerical experiments on checkerboard patches, corner-singularity refinement, and an inductor machine demonstrate the method’s efficacy and guide practical strategies to maintain p-robust performance in the presence of non-matching interfaces and coefficient jumps. The results confirm that a nested trace space approach, combined with appropriate primal DOFs at T-junctions, yields scalable, robust solvers suitable for large-scale IgA applications with adaptive refinement. Theoretical and numerical findings indicate that the proposed framework preserves optimal convergence rates while enabling efficient parallel computation for complex multi-patch geometries.
Abstract
Isogeometric Analysis is a variant of the finite element method, where spline functions are used for the representation of both the geometry and the solution. Splines, particularly those with higher degree, achieve their full approximation power only if the solution is sufficiently regular. Since solutions are usually not regular everywhere, adaptive refinement is essential. Recently, a multi-patch-based adaptive refinement strategy based on recursive patch splitting has been proposed, which naturally generates hierarchical, non-matching multi-patch configurations with T-junctions, but preserves the tensor-product structure within each patch. In this work, we investigate the application of the dual-primal Isogeometric Tearing and Interconnecting method (IETI-DP) to such adaptive multi-patch geometries. We provide sufficient conditions for the solvability of the local problems and propose a preconditioner for the overall iterative solver. We establish a condition number bound that coincides with the bound previously shown for the fully matching case. Numerical experiments confirm the theoretical findings and demonstrate the efficiency of the proposed approach in adaptive refinement scenarios.