On a Hybrid Mixed Domain Decomposition Method

2026-04-24

AI Summary

This work proposes a novel framework inspired by hybridized discontinuous Galerkin methods to address the insufficient stability and accuracy of hybrid domain decomposition approaches in coupling interface fluxes and variables. By reformulating the divergence operator in the variational form as an $L^2$ quantity both within subdomains and on interfaces—and introducing a stabilization term—the method innovatively replaces traditional interface trace functions with $L^2$ distributions, preserving variational consistency while enabling an explicit, uniformly bounded error analysis with respect to the stabilization parameter $\tau$. The dual variable is discretized using Raviart–Thomas elements, while primal and hybrid variables are approximated by piecewise polynomials, yielding a consistent perturbed Galerkin numerical scheme. Numerical experiments on curved quadrilateral meshes demonstrate $(q+1)$-order convergence for primal and hybrid variables, $(q+1/2)$-order convergence for the dual variable, and robustness of errors with respect to $\tau$.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present a domain decomposition formulation based on hybridization which is inspired by hybridized discontinuous Galerkin (HDG) methods, that enhance mixed domain decomposition methods by incorporating stabilization terms. Unlike discontinuous Galerkin methods, our analysis of the proposed finite element method is based on a corresponding consistent variational formulation and a perturbed Galerkin method. In the variational formulation the divergence appears not only within subdomains, but also as an $L^2$-surface quantity on the interfaces. Furthermore, the traces of the finite element functions on the interfaces are replaced by $L^2$-distributions. The well-posedness of the perturbed Galerkin method is shown for an appropriate choice of subspaces, in a manner similar to that of the variational formulation. For the finite element method we use Raviart-Thomas elements for the dual variable and piecewise polynomials for the primal and hybrid variables, respectively. We perform an analysis of the discretization error which is explicit in the stabilization parameter $τ$. Numerical experiments for piecewise smooth solutions using finite element spaces of order~$q$ on curved quadrilateral meshes confirm the predicted convergence rate of $q+1$ for small values of $τ$. In the error analysis we observe the discretization error to be uniformly bounded in $τ$. Even for large $τ$ values the observed convergence rates for the primal and for the hybrid variables are $q+1$. For the dual variable the convergence rate depends on the stabilization parameter and the mesh-width, with an asymptotic rate of $q+\tfrac12$.

Problem

• domain decomposition
• mixed methods
• stabilization
• hybridization
• finite element method

Innovation

• hybridized domain decomposition
• perturbed Galerkin method
• stabilization parameter
• Raviart-Thomas elements
• L²-interface traces