Analysis of BDDC preconditioners for non-conforming polytopal hybrid discretisation methods

Abstract: In this work, we build on the discrete trace theory developed in [Badia, Droniou, Tushar, arXiv (2024)] to analyse the rate of convergence of the Balancing Domain Decomposition by Constraints (BDDC) preconditioner generated from non-conforming polytopal hybrid discretisation. We prove polylogarithmic condition number bounds for the preconditioner that are independent of the mesh parameter and number of subdomains, and hold on polytopal meshes. The analysis hinges on the continuity of a face truncation operator, which we prove in the fully discrete polytopal setting. To validate the theory, we present two numerical experiments: the first verifies the truncation estimate, and the second -- a weak scalability test -- verifies the robustness of the condition number bounds for BDDC when applied to second-order elliptic problems discretised using discontinuous skeletal methods, specifically Hybridizable Discontinuous Galerkin (HDG) and Hybrid High-Order (HHO) methods.