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

TL;DR

This work extends discrete trace theory to non-conforming polytopal hybrid discretisations and analyzes the Balancing Domain Decomposition by Constraints (BDDC) preconditioner for such systems. It introduces a continuity framework for a face truncation operator and proves polylogarithmic, mesh- and subdomain-count-independent condition-number bounds on polytopal meshes. The authors provide a rigorous functional-analytic setting for BDDC in hybrid spaces and validate the theory through comprehensive numerical experiments on HDG and HHO methods, including weak scalability and robustness to large coefficient jumps. The results demonstrate that the proposed BDDC preconditioner is robust, scalable, and effective for discontinuous skeletal methods on general polytopal meshes. The work advances practical solvers for modern polytopal discretisations and supports their use in large-scale simulations via GridapSolvers.jl.

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.

Figures (10)

• Figure 1: Layers of faces from the boundary of $\Gamma$.
• Figure 2: Illustration of the uniform cone condition.
• Figure 3: Visual representation of the sets $\mathcal{C}_f$ (full cone), $\mathcal{C}_{fd_0}$ (truncated cone), and the set discussed in Lemma \ref{['lem:bddc.Cfd']}-\ref{['lem:bddc.Cfd.card']}. Note that while these sets are unions of cells, the figure shows a simplified geometric approximation of the domain they cover.
• Figure 4: Visual representation of the spaces in the BDDC framework. Spaces are represented for order $k = 0$ on a $4 \times 4$ cartesian mesh divided into four $2 \times 2$ subdomains. Interior and interface face DoFs are drawn as black and red dots respectively. Coarse DoFs (constraints) are represented as blue squares. Red zeroes represent the vanishing DoFs (fixed zero value) in the bubble space. Cell DoFs are not present for clarity. From left to right: sub-assembled space $\ul{U}_{h}$, BDDC space $\ul{\widetilde{U}}_h$, global hybrid space $\widehat{\ul{U}}_{h}$, bubble space $\ul{U}_{h,0}$.
• Figure 5: Schematic representation of the operator $\mathcal{Q}_h$. Interior and interface face DoFs are drawn as black and red dots respectively. Cell DoFs are not present for clarity.

Theorems & Definitions (18)

• Theorem 1.2: Trace inequality
• Theorem 1.3: Lifting
• Theorem 2.2: Estimate of truncated boundary function
• Remark 2.3: Scaling
• Proposition 2.4
• Lemma 2.5: Cardinality bound for $\mathcal{I}_{ft}^m$
• proof
• Lemma 2.6: Volume and Cardinality Estimates for $\mathcal{C}_{f d_0}$
• proof
• Lemma 2.7: Weighted sum bound for interior faces