Mixed Formulation of the Hodge Laplace Problem

• The mixed formulation decouples the Hodge Laplace problem into a saddle-point system, separating the primal variable, its derivative, and harmonic constraints.
• Adaptive finite element methods and finite element exterior calculus provide robust error estimation and optimal convergence on complex geometries.
• Local coderivative techniques and mass-lumped inner products enhance computational efficiency in simulating elliptic and multiphysics problems.

The mixed formulation of the Hodge Laplace problem is a framework central to the analysis and numerical approximation of partial differential equations involving differential forms, especially on manifolds and domains with complex geometry or topology. This formulation plays a foundational role in finite element exterior calculus (FEEC) and underpins adaptive and structure-preserving algorithms for elliptic problems on surfaces, in domains with nontrivial harmonic fields, and in multiphysics and coupled PDE systems.

1. Foundations of the Mixed Formulation

The classical Hodge Laplacian is the second-order elliptic operator acting on differential k-forms, given by

$L = d d^* + d^* d,$

where $d$ is the exterior derivative and $d^*$ its formal adjoint with respect to the $L^2$ inner product. The associated strong formulation is to find $u$ such that $L u = f$, but this direct approach is challenging when the domain has nontrivial topology (i.e., nonzero harmonic forms) or when designing numerical methods, due to difficulties with constructing finite element spaces for $V^k \cap V^*_k$.

The mixed formulation circumvents these issues by introducing auxiliary variables to decouple the primal field, its derivative, and the harmonic projection, resulting in a saddle-point system. In a Hilbert complex setting, the mixed formulation seeks a triple $$(\sigma, u, p) \in V^{k-1} \times V^k \times \mathfrak{H}^k$$ such that

$$\begin{aligned} \langle \sigma, \tau \rangle - \langle d\tau, u \rangle & = 0, && \forall\, \tau \in V^{k-1}, \ \langle d \sigma, v \rangle + \langle d u, dv \rangle + \langle p, v \rangle & = \langle f, v \rangle, && \forall\, v \in V^k, \ \langle u, q \rangle & = 0, && \forall\, q \in \mathfrak{H}^k, \end{aligned}$$

where $\mathfrak{H}^k$ is the space of harmonic forms (ker$\,d\, \cap$ ker$\,d^*$) (Holst et al., 2014). This directly addresses the existence of nontrivial harmonic components and avoids the construction of finite element spaces approximating $V^k \cap V^*_k$.

2. Adaptive Finite Element Methods

Adaptive finite element methods (AFEM) for the mixed Hodge Laplacian are designed to combine a posteriori error estimation, mesh refinement, and structure-preserving finite elements. The paradigm is based on the loop: $$\text{SOLVE} \to \text{ESTIMATE} \to \text{MARK} \to \text{REFINE}$$ Details of this process for hypersurfaces involve:

• SOLVE: On a given triangulation (possibly of a polygonal approximant to a smooth manifold), compute $(\sigma_h, u_h, p_h)$ solving the discrete mixed problem.
• ESTIMATE: Compute a posteriori error indicators $\eta_T$ for each element $T$, incorporating data oscillation terms via mappings between the true surface $M$ and the mesh $M_A$ with operators like $i_A$ and $\pi_A$.
• MARK: Use Dörfler marking or similar strategies to select elements for refinement.
• REFINE: Refine the selected elements, ensuring mesh regularity.

The analysis provides contraction results for combined energy, estimator, and oscillation quantities, and proves that if the true solution is in approximation class $\mathcal{A}^s$, the error decays at the optimal $N^{-s}$ rate (Holst et al., 2014). The AFEMs for the mixed Hodge Laplacian rely crucially on bounded cochain projections and the FEEC property of forming subcomplexes that preserve commutativity with $d$.

3. Finite Element Exterior Calculus (FEEC) and Discrete Complexes

FEEC is the natural habitat for mixed Hodge Laplacian formulations, providing both analytical and algebraic structure for numerical schemes. In FEEC, the continuous de Rham complex

$$0 \rightarrow H\Lambda^0(\Omega) \xrightarrow{d} H\Lambda^1(\Omega) \xrightarrow{d} \cdots \xrightarrow{d} H\Lambda^n(\Omega) \rightarrow 0$$

is approximated by discrete subcomplexes of finite element spaces $\mathcal{P}_r\Lambda^k$ or $\mathcal{P}_r^-\Lambda^k$ with the exterior derivative $d$ mapping one discrete space to the next (Holst et al., 2014). Bounded cochain projections (such as $I_h$, $\pi_h$, or $\Pi_h$) with $d\,I_h = I_h\,d$ allow for crucial commuting diagram properties.

FEEC methods extend to contexts where the discrete complex is not precisely a subcomplex of the continuous one (variational crimes), particularly when the mesh is a polyhedral approximation to a curved surface—push-forward and pull-back operators ($i_A$, $\pi_A$) mediate between the discrete and continuous complexes (Holst et al., 2014). Such generalizations are essential for robust PDE approximation on curved or evolving manifolds.

4. Local and Nonlocal Coderivatives

A persistent issue in the discrete mixed formulation is the nonlocality of the discrete coderivative $d_h^*$. While $d$ is local, the standard FEEC methods define $d_h^*$ implicitly via

$(d_h^* u_h, \tau_h) = (u_h, d\tau_h),$

which typically requires inversion of a global mass matrix on $V_h^{k-1}$, producing nonlocal effects (Lee et al., 2016).

To address this, methods modify the inner product on the $V_h^{k-1}$ space, using quadrature rules or mass-lumping to create block-diagonal (and thus local) mass matrices, as in

$$(u, v)_h = \sum_{T} \frac{1}{n+1}|T| \sum_{x \in \Delta_0(T)} (u(x), v(x))_{\text{Alt}},$$

where $\Delta_0(T)$ labels the simplex vertices (Lee et al., 2016, Lee, 2019). Higher-order, mass-lumped, or cubical element designs yield discrete coderivatives that are local in the mesh, improving computational efficiency and conformity to physical locality (important in Darcy, Maxwell, and related systems).

5. Harmonic Fields and Constraints

Harmonic forms are the kernel of both $d$ and $d^*$. The mixed formulation must account for the possible nontriviality of $\mathfrak{H}^k$, imposing the orthogonality condition

$$\langle u, q\rangle = 0, \qquad \forall q \in \mathfrak{H}^k,$$

to ensure uniqueness. Discrete approximations may require construction of discrete harmonic spaces $S_h$ and appropriate constraints in the variational system (Holst et al., 2014, Lee et al., 2016).

For problems without harmonic fields (e.g., simply connected domains or certain boundary conditions), analyses focusing on only the $(\sigma,u)$ variables yield simplified adaptive and error estimates (Chen et al., 2016).

6. Adaptive Convergence and Optimality

The mixed formulation in the FEEC framework admits a posteriori analysis. Residual-type estimators are shown to be both reliable and efficient, bounding the error in the $(\sigma, u, p)$ variables. Quasi-orthogonality allows contraction arguments similar to those of standard scalar elliptic AFEM (Holst et al., 2014, Chen et al., 2016). Marking strategies that separately track different error components (as in marking both $u$ and $\sigma$) are shown to be necessary for robust convergence in saddle-point problems.

When the data and geometry possess further regularity, the convergence rates can be shown to be optimal and uniform over the refinement iterations, even in the absence of mesh-size restrictions or small initial meshes (Chen et al., 2016). The optimality is always referenced to the approximation class of the true solution.

7. Applications and Computational Practice

Mixed formulations of the Hodge Laplace problem are the computational basis for:

• Surface finite element methods (SFEM) for the Laplace–Beltrami equation (Holst et al., 2014);
• Electromagnetic, Darcy, and elasticity-type PDEs involving de Rham complexes;
• Simulations on surfaces or embedded manifolds (computer graphics, materials science);
• Problems where adaptivity enables computational efficiency on complex geometries.

Adaptive mixed FEEC methods, with robust and efficient error estimators and local coderivative operators, are now routinely used for high-fidelity surface PDE simulations and are a cornerstone for future PDE discretization research (Holst et al., 2014, Lee et al., 2016, Chen et al., 2016).

Table: Mixed Formulation Components

Variable	Space	Role
$\sigma$	$V^{k-1}$	Flux / auxiliary field, $d^*u$ proxy
$u$	$V^{k}$	Primal unknown (differential form)
$p$	$\mathfrak{H}^{k}$	Harmonic constraint variable

8. Extensions and Future Directions

Current research is pushing these ideas towards:

• High-order, mass-lumped, or cubical finite elements for local coderivative computation (Lee, 2019);
• Broken or nonconforming FEEC schemes, e.g., CONGA methods for increased locality and efficient parallelism (Campos-Pinto et al., 2021);
• Primal-only (single-field) schemes exploiting discrete Poincaré–Lefschetz duality with robust convergence on complex domains (Zhang, 2022);
• Preconditioner development that is robust across parameters and mesh scales via parameter-dependent fitted norms in the saddle-point mixed context (Boon et al., 31 Jul 2025).

Conclusion

The mixed formulation of the Hodge Laplace problem, especially when treated via FEEC, AFEM, and local discrete operator design, provides a structurally sound, numerically efficient, and theoretically robust framework for the approximation of elliptic PDEs on manifolds and domains with nontrivial topology. By abstracting away the need for constructing spaces on complicated intersections and by incorporating harmonic constraints, the mixed approach enables convergence and optimality analyses and supports the development of adaptive and high-order methods applicable to a wide range of geometric and physical problems (Holst et al., 2014, Lee et al., 2016, Chen et al., 2016, Lee, 2019).