Hybrid High-Order Method

• Hybrid High-Order (HHO) methods are advanced numerical schemes that use local polynomial approximations and interface reconstructions to accurately solve PDEs.
• They combine gradient reconstruction with high-order stabilization to ensure robust, locally conservative, and efficient discretizations on arbitrary polytopal meshes.
• The approach is validated through convergence analysis, discrete Sobolev embeddings, and numerical experiments for nonlinear and degenerate elliptic problems.

A Hybrid High-Order (HHO) method is a class of numerical discretizations for partial differential equations (PDEs) that leverages approximating polynomial spaces both inside mesh elements and on their interfaces (faces), together with local reconstruction operators and stabilization terms. Developed within the context of polytopal mesh flexibility and arbitrary approximation order, HHO methods offer robust, high-order, and locally conservative schemes that combine features of discontinuous Galerkin and hybridizable approaches, while facilitating efficient static condensation. The HHO framework is particularly effective for nonlinear and possibly degenerate elliptic PDEs, as demonstrated in applications to steady non-linear Leray–Lions problems, and offers discrete functional analysis tools of broad interest.

1. Core Principles of the Hybrid High-Order Method

The fundamental structure of HHO methods is defined by two main ingredients acting locally on mesh elements:

• Gradient reconstruction: For each mesh element, a local operator is constructed to recover a high-order approximation of the gradient of the solution from the hybrid set of discrete unknowns (which includes both element and face polynomials).
• High-order stabilization: A properly designed stabilization term generalizes earlier approaches in the linear context, weakly enforcing consistency between interface unknowns and the reconstructed polynomial inside each element.

The discrete unknowns are organized such that for a given polynomial degree $k$, local spaces consist of:

• Element-based polynomials (typically of degree $k$ or higher).
• Face-based polynomials (often of degree $k$, attached to each boundary face of the element).

The high-order stabilization penalizes the discrepancy (in a projected sense) between element and face unknowns, ensuring both stability and optimal approximation even on general polytopal meshes.

2. Discrete Functional Analysis and Sobolev Embeddings

A key analytical advance underlying the HHO methodology is the development of discrete functional analysis tools tailored to local polynomial and hybrid spaces. In the context of nonlinear Leray–Lions problems, the discrete Sobolev embeddings take the following form. For any homogeneous Dirichlet discrete function $v_h$, with element-based “potential” $u_h$, the following inequality is established:

$\|u_h\|_{L^p(\Omega)} \leq C \|v_h\|_{1,p,h},$

where the discrete $W^{1, p}$-“norm” $\|v_h\|_{1,p,h}$ is defined element-wise, consisting of the $L^p$-norm of the gradient of the reconstructed potential and face penalty terms measuring jumps between element and face degrees of freedom. The exponent in the embedding can be any $q$ in $[1, p^*]$, with $p^*$ the usual Sobolev exponent.

Choosing $q=p$ recovers a discrete Poincaré–Wirtinger–Sobolev inequality:

$\|u_h\|_{L^p(\Omega)} \leq C \|v_h\|_{1,p,h},$

which is fundamental for establishing uniform a priori estimates, compactness via a Rellich–Kondrachov type theorem, and ultimately convergence of the discrete solution to the continuous one (Pietro et al., 2015).

3. Convergence, Compactness, and Discrete Compact Embeddings

The convergence proof for HHO methods applied to the Leray–Lions nonlinear elliptic equations employs a compactness technique adapted to the hybrid setting. The main steps are:

• The discrete Poincaré–Wirtinger–Sobolev inequality enables uniform boundedness of the sequence of element-based potentials in $L^p$.
• Combined with control over the reconstructed gradients and penalties on jumps, this yields precompactness in suitable discrete spaces.
• Through a limit-passage and monotonicity arguments suitable for $p$-structure nonlinearities, it is shown that the discrete solution sequence converges (up to a subsequence) in measure to a solution of the continuous problem.

This approach necessitated the development of a family of discrete functional analysis tools, including:

• Direct and reverse Lebesgue and Sobolev embeddings for local polynomial spaces.
• $L^p$-stability and $W^{s,p}$-approximation properties for $L^2$-projectors.
• Discrete Sobolev embeddings for hybrid polynomial and broken spaces.

These results are nontrivial extensions of their continuous counterparts and have implications that span beyond the specific model problem.

4. Methodology for Implementation

In standard implementation, the following workflow is prescribed:

1. Construction of Local Spaces: For each mesh element, associate element and face polynomial unknowns of chosen degree, ensuring compatibility with general polytopal element shapes.
2. Gradient Reconstruction: Locally solve for a reconstruction of the potential (the “gradient recovery”), which serves as a high-order polynomial in each element.
3. High-Order Stabilization: Incorporate a face-based stabilization term, which penalizes (in an $L^2$-projected sense) the difference between face and element values, thus controlling the nonconformity of the discrete approximation.
4. Assembly of the Global System: Compose the local contributions into the global variational formulation, potentially enabling static condensation to eliminate element degrees of freedom.
5. Solve Nonlinear System: Employ Newton or other suitable nonlinear solvers, leveraging the compactness and stability properties proved analytically.

5. Performance, Flexibility, and Mesh Compatibility

Key features of the HHO method as presented for Leray–Lions problems include:

• Support for Arbitrary Order: The scheme is provably stable and convergent for any polynomial order $k$, allowing selection tailored to the expected regularity of the solution or desired accuracy.
• Mesh Generality: The method is formulated on general polytopal meshes, accommodating nonmatching interfaces, hanging nodes, and irregular element geometries.
• Static Condensation: The algebraic structure enables elimination of local, element-based unknowns, leading to a global linear system that is coupled only through the mesh skeleton (faces).
• Robustness for Nonlinear Problems: The discrete Poincaré–Wirtinger–Sobolev inequality, together with the developed compactness framework, provides robustness for strongly nonlinear and potentially degenerate elliptic equations.

A table summarizing the main properties is as follows:

Feature	Description	Reference
Discrete Sobolev Embedding	$\|u_h\|_{L^p(\Omega)} \leq C \|v_h\|_{1,p,h}$ for any $v_h$ in hybrid space	(Pietro et al., 2015)
Mesh Type	Arbitrary polytopal meshes, with or without matching interfaces	(Pietro et al., 2015)
Approximation Order	Any $k\geq 0$	(Pietro et al., 2015)
Static Condensation	Local elimination of element degrees of freedom (optional)	(Pietro et al., 2015)
Nonlinear Robustness	Convergence proof via compactness, uniform stability by discrete inequalities	(Pietro et al., 2015)

6. Numerical Experiments and Validation

Numerical results presented in (Pietro et al., 2015) exhibit:

• Verification of the a priori estimates and discrete compactness on a set of canonical test problems.
• Empirical demonstration that the convergence rates match the theoretical predictions, confirming the optimality of the design for both mesh and order flexibility.
• Validation that the method performs robustly for nonlinear and degenerate equations and on non-standard (e.g., polygonal) mesh families.

The tests further confirm that the discrete Poincaré–Wirtinger–Sobolev inequality underpins the observed uniform stability of the approximation and the attainment of expected error orders.

7. Significance and Further Developments

The introduction and rigorous analysis of HHO methods for Leray–Lions and related nonlinear elliptic PDEs marks a significant development in the numerical analysis of high-order and hybridizable schemes. The discrete functional analysis tools developed for this context—particularly the embedding and compactness results—are of general value for other hybrid and discontinuous methods, or for problems where direct application of classical (continuous) results is not possible due to nonconforming or hybrid discretizations. The methodology has influenced subsequent extensions to parabolic, mixed-form, and higher-order PDEs, as well as adaptive algorithms that require such discrete analytic underpinnings.

The approach represents a unified, high-order, and robust methodology for the approximation of nonlinear elliptic problems on general meshes, with discrete stability and compactness ensured by functional inequalities tailored to hybrid polynomial spaces (Pietro et al., 2015).