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

TL;DR

This paper analyzes the convergence of BDDC preconditioners for non-conforming polytopal hybrid discretizations, providing polylogarithmic bounds on the condition number that are mesh- and subdomain-independent, validated through numerical experiments.

Contribution

It establishes polylogarithmic bounds on the BDDC preconditioner condition number for non-conforming polytopal discretizations, extending the theory to fully discrete settings and validating with numerical tests.

Findings

Polylogarithmic bounds on condition number independent of mesh and subdomains.

Validation of bounds through numerical experiments with HDG and HHO methods.

Preconditioner robustness for problems with large coefficient jumps.

Abstract

In this work, we build on the discrete trace theory developed by Badia, Droniou, and Tushar (Foundations of Computational Mathematics, in press, 2025; \href{https://doi.org/10.1007/s10208-025-09734-6}{doi:10.1007/s10208-025-09734-6}) to analyze the convergence rate of the Balancing Domain Decomposition by Constraints (BDDC) preconditioner generated from non-conforming polytopal hybrid discretizations. We prove polylogarithmic bounds on the condition number for the preconditioner that are independent of the mesh parameter and the number of subdomains, and that hold on polytopal meshes. The analysis relies on the continuity of a face truncation operator, which we establish in the fully discrete polytopal setting. To validate the theory, we present numerical experiments that confirm the truncation estimate and condition number bounds. In particular, we conduct weak scalability tests for second-order elliptic problems discretized using discontinuous skeletal methods, specifically Hybridizable Discontinuous Galerkin (HDG) and Hybrid High-Order (HHO) methods. We also demonstrate the robustness of the preconditioner for piecewise discontinuous coefficients with large jumps.