On a Hybrid Mixed Domain Decomposition Method

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 $q+\tfrac12$.

• The paper's main contribution is the development and analysis of a Hybrid Mixed Domain Decomposition method that uses stabilization terms to enable explicit domain coupling.
• It rigorously proves well-posedness and establishes uniform error bounds in τ, with optimal convergence rates of order h^(q+1) for the primal and hybrid variables.
• Numerical experiments validate the approach on Poisson’s equation, demonstrating robust performance on both matching and non-matching meshes with variable coefficients.

Hybrid Mixed Domain Decomposition: A Variational and Numerical Analysis

Problem Setting and Motivation

The paper "On a Hybrid Mixed Domain Decomposition Method" (2604.22543) develops and analyzes a finite element domain decomposition approach for elliptic PDEs, leveraging advances from hybridized discontinuous Galerkin (HDG) methods. The central motivation lies in the interface treatment between non-overlapping subdomains—but instead of classical approaches, the Hybrid Mixed Domain Decomposition (HMDD) method incorporates stabilization terms à la Lehrenfeld-Schöberl, extending the classical mixed formulation by enabling explicit domain coupling. This allows for both optimal convergence and the possibility of applying the method to matching or non-matching meshes and variable coefficients.

The work's focus is on Poisson’s equation with homogeneous Dirichlet boundary data, re-cast in a mixed form using a flux variable. The formulation aims to combine the reduced global coupling, positive definiteness, and efficient Schur-complement properties of hybridized schemes with the flexibility and rigor of mixed finite elements.

Formulation of the Method

Mixed Hybrid Formulation

The computational domain $\Omega$ is partitioned into non-overlapping subdomains. The interface $\Gamma$ between neighboring subdomains (the skeleton) plays a key role. The hybrid variable $\mu$ represents the trace of the primal variable $u$ on $\Gamma$ (Figure 1).

Figure 1: Exemplary depiction of the domain $\Omega$ decomposed into four disjoint subdomains. Green lines denote the interface $\Gamma$; blue arrows are arbitrary but fixed normal directions.

Inside each patch, a standard mixed formulation is used: find $(\mathbf{q}, u, \mu)$ such that

$$\begin{aligned} \kappa^{-1}\mathbf{q} - \nabla u &= 0 \text{ in }\Omega\setminus\Gamma, \ -\text{div}~\mathbf{q} &= f \text{ in }\Omega\setminus\Gamma, \ \end{aligned}$$

and coupling on $\Gamma$ is imposed weakly via stabilization and hybrid variables.

A key feature is the use of stabilization parameter $\Gamma$0, controlling the strength of penalty on primal trace jumps across $\Gamma$1. This recovers classical methods (e.g., Raviart-Thomas hybrid mixed as $\Gamma$2 and CG as $\Gamma$3), but for $\Gamma$4, it enables non-conforming meshing.

Variational and Discrete Setting

The authors provide a rigorous variational formulation, using duality products and carefully constructed function spaces (notably, $\Gamma$5 for the primal variable, and $\Gamma$6 for the flux). These spaces allow for $\Gamma$7-based distributions both in the subdomain interiors and on the skeleton, weighted appropriately by $\Gamma$8. The norm and decomposition structure ensure robust inf-sup stability independent of the choice of $\Gamma$9.

In the finite element setting, subdomain spaces use Raviart-Thomas elements for the flux, standard polynomials for $\mu$0 and the hybrid variable, and approximate traces on $\mu$1 are handled via $\mu$2 projection.

Figure 2: Computational domain $\mu$3 and reference solution component $\mu$4.

Figure 3: Numerical solution $\mu$5 and $\mu$6 (red lines) at polynomial order $\mu$7 (upper) and $\mu$8 (lower) for increasing mesh refinement.

Theoretical Analysis

Well-Posedness

A major theoretical achievement is the proof of well-posedness uniformly in $\mu$9. Through a detailed inf-sup theory, the authors show that the coupled system is stable for all $u$0, matching mixed finite element stability arguments. The analysis extends naturally to high-order finite elements and curved meshes.

Error Analysis

The discretization error is bounded explicitly in terms of $u$1 (mesh size), $u$2 (polynomial order), $u$3, and solution regularity: $u$4 for sufficiently regular $u$5. Similar error bounds are established for flux and divergence errors, demonstrating that for moderate $u$6, optimal $u$7 convergence rates are observed—independent of the stabilization.

Notably, the error is uniformly bounded in $u$8, and observed rates for the primal and hybrid variables remain $u$9 even for large $\Gamma$0. For the flux variable, the rate is $\Gamma$1 asymptotically as $\Gamma$2 or $\Gamma$3 varies.

Numerical Experiments

A comprehensive numerical study validates the theoretical predictions. The test case features a radially symmetric domain with an interface at the unit circle and material jump. Polynomial orders $\Gamma$4 and $\Gamma$5 were tested across several mesh resolutions and a wide range of stabilization parameters $\Gamma$6.

Figure 4: $\Gamma$7 error in $\Gamma$8 as a function of $\Gamma$9 for order $\Omega$0.

Figure 5: Error plots for $\Omega$1, $\Omega$2, and $\Omega$3 across $\Omega$4, as $\Omega$5 varies for $\Omega$6.

Figure 6: Error plots for the flux variable on $\Omega$7: normal component $\Omega$8, primal jump $\Omega$9, and hybrid error, for $\Gamma$0 and different $\Gamma$1.

Key empirical findings:

• For fixed $\Gamma$2, all error measures remain bounded for large $\Gamma$3, confirming independence from stabilization magnitude.
• Both $\Gamma$4 and $\Gamma$5 consistently show optimal convergence rates of $\Gamma$6 with respect to $\Gamma$7, robust to $\Gamma$8.
• The flux and its divergence show intermediate rates that transition depending on $\Gamma$9 and $(\mathbf{q}, u, \mu)$0—matching analysis.

The authors systematically investigate both $(\mathbf{q}, u, \mu)$1- and $(\mathbf{q}, u, \mu)$2-asymptotic regimes, clarifying convergence behavior in all relevant parameter limits.

Implications and Future Directions

This work establishes a robust theoretical and numerical foundation for HMDD as a versatile domain decomposition approach for mixed formulations. The flexibility of $(\mathbf{q}, u, \mu)$3 enables straightforward application to non-matching grids (mortar methods), higher dimensions, and more general PDEs (e.g., Maxwell or Helmholtz equations).

Potential directions include:

• Extending the variational analysis to general non-matching grids and multipatch isogeometric discretizations.
• Exploiting the positive-definite, sparsified Schur complement structure for efficient solver design.
• Applying the framework to more complex coupled field problems and time-dependent PDEs.

The variational underpinning also opens the door to rigorous mortar-like coupling of multi-physics and multi-mesh simulations.

Conclusion

The HMDD method represents a theoretically well-founded, high-order accurate approach to domain decomposition for mixed formulations of elliptic problems, unifying and extending previous hybrid and hybridized Galerkin ideas. Its robustness with respect to stabilization, rigorous error analysis, and empirical validation set the stage for broad applicability in large-scale and geometrically complex PDE simulations.