Gradient Discretisation Method (GDM)

• Gradient Discretisation Method (GDM) is a unified abstract framework that replaces continuous PDE operators with discrete analogues to ensure convergence and error control.
• It integrates various spatial discretisations—including finite elements, finite volumes, and mimetic finite differences—for applications like optimal control and variational inequalities.
• GDM offers robust error estimates and superconvergence results, making it a vital tool for verifying and advancing numerical schemes in complex PDE simulations.

The Gradient Discretisation Method (GDM) is a general, abstract framework for analyzing and implementing numerical schemes for partial differential equations and variational problems, unifying the treatment of diverse spatial discretisations such as conforming and non-conforming finite elements, mixed and hybrid methods, finite volumes, and mimetic finite differences. Initially formulated to systematize and extend convergence results for diffusion PDEs, GDM has grown into a major tool, distinguished by its strong abstraction and its ability to encompass new, nonstandard, or hybrid numerical methods under a single analytical umbrella.

1. Core Principles and Abstract Structure

GDM is based on the replacement of the continuous operators and function spaces in the weak formulation of a PDE or variational problem by discrete analogues—called a gradient discretisation (GD)—which must satisfy a set of minimal qualitative properties (coercivity, consistency, limit-conformity, and compactness).

A typical GD for a second-order elliptic or parabolic problem consists of:

• $X_{\mathcal{D},0}$: A finite-dimensional space of discrete unknowns.
• $\Pi_{\mathcal{D}}$: A function reconstruction operator, mapping discrete unknowns to $L^2$ functions.
• $\nabla_{\mathcal{D}}$: A discrete gradient operator, mapping discrete unknowns to vector fields in $L^2(\Omega)^d$.
• For some problems, further trace ($\mathbb{T}_{\mathcal{D}}$) or interpolation ($\mathcal{I}_{\mathcal{D}}$) operators are defined.

A gradient scheme (GS) is the resulting numerical method obtained by substituting the continuous function and gradient in the weak formulation by the reconstructed quantities from GD. For time-dependent or more complex problems, the GD is extended to incorporate time discretisation and, if necessary, additional structure for handling boundary or initial data.

The abstract nature of GDM enables simultaneous analysis of widely varied numerical methods—provided the reconstructions and discrete spaces satisfy the GDM properties, error and convergence results transfer immediately.

2. Application to Optimal Control and Variational Inequalities

GDM has been successfully applied to a wide spectrum of problems, including optimal control of diffusion equations, nonlinear variational inequalities, and models with complex boundary or coupling conditions.

For optimal control problems governed by diffusion processes, the GDM is used to discretise the optimality system (KKT system), simultaneously covering state, adjoint, and control variables. For the Dirichlet-constrained tracking problem,

$$\min_{u \in \mathcal{U}_{ad}} J(y, u) = \frac{1}{2}\| y - y_d \|_{L^2(\Omega)}^2 + \frac{\alpha}{2}\| u-u_d \|_{L^2(\Omega)}^2$$

subject to

$$-(A\nabla y) = f + u \quad \text{in } \Omega, \quad y = 0 \text{ on } \partial\Omega,$$

the GDM provides a discrete system mirroring the continuous optimality conditions: $$\begin{cases} a_\mathcal{D}(y_\mathcal{D}, w_\mathcal{D}) = (f + u_h, \Pi_\mathcal{D} w_\mathcal{D}),\;\forall w_\mathcal{D}\ a_\mathcal{D}(w_\mathcal{D}, p_\mathcal{D}) = (\Pi_\mathcal{D} y_\mathcal{D} - y_d, \Pi_\mathcal{D} w_\mathcal{D}),\;\forall w_\mathcal{D}\ (\Pi_\mathcal{D} p_\mathcal{D} + \alpha(u_h-u_d), v_h-u_h) \geq 0,\;\forall v_h \in \mathcal{U}_h \end{cases}$$ with $$a_\mathcal{D}(w, v) = \int_\Omega A \nabla_\mathcal{D} w \cdot \nabla_\mathcal{D} v \, dx$$.

For variational inequalities, including obstacle and Signorini problems, GDM is used to construct consistent discretisations of the inequality constraints, ensuring that the discrete counterpart of the convex feasible set is preserved and that convergence to the weak continuous solution occurs under general mesh assumptions.

3. Error Estimates and Superconvergence

A central achievement of the GDM framework is the derivation of robust, unified a priori error estimates, often encompassing cases not previously accessible. For elliptic control and variational inequality problems, the following generic GDM error estimate is established: $$\| \Pi_\mathcal{D} \psi_\mathcal{D} - \psi \| + \| \nabla_\mathcal{D} \psi_\mathcal{D} - \nabla \psi \| \lesssim S_\mathcal{D}(\psi) + W_\mathcal{D}(A\nabla \psi),$$ where the terms $S_\mathcal{D}$ (interpolation/consistency) and $W_\mathcal{D}$ (limit-conformity) are determined by the mesh and discretisation scheme.

A particularly notable result is the established superconvergence for post-processed controls:

• On quasi-uniform meshes and under standard regularity,

$$\| \widetilde{u} - \widetilde{u}_h \|_{L^2} \lesssim h^2$$

where $\widetilde{u}_h$ denotes the post-processed discrete control.

• For general meshes,

$$\| \widetilde{u} - \widetilde{u}_h \|_{L^2} \leq C h^{2-\frac{1}{2^*}}$$

These rates extend classical superconvergence to non-conforming $\mathbb{P}_1$ finite elements and hybrid/mimetic schemes (HMM, hMFD), and are the first such results in the literature for these methods.

For nonlinear variational inequalities and nonconforming or hybrid schemes, similar strong convergence results are obtained for the approximated solution and the discrete gradient, provided the standard GDM properties hold.

4. Practical Implementation and Numerical Validation

The abstraction of GDM is matched by practical versatility. Implementation proceeds by selecting a concrete discretisation (for example, conforming or non-conforming finite elements, finite volumes, hybrid mimetic mixed methods), then specifying the corresponding GD: how are functions and gradients reconstructed from the scheme’s degrees of freedom? Provided these reconstructions respect the formal definitions given by GDM, the convergence theorems apply.

In practice:

• For non-conforming finite elements (e.g., Crouzeix-Raviart), GDM defines broken gradient and function reconstructions that adhere to the scheme’s local nature.
• In the HMM/mimetic context, cell/face unknowns and their reconstructions are handled by tailored operators within the GDM framework, permitting application on highly irregular or polytopal meshes.

Numerical experiments across multiple works demonstrate that GDM-based methods, for both control and constraint problems, realize the anticipated error orders, including superconvergent rates for post-processed variables and resilience to mesh distortion or irregularity.

For example, for Dirichlet control problems in convex domains, both conforming and non-conforming finite elements as well as mimetic methods yield:

• Linear convergence for original control variable, quadratic for post-processed control;
• Clear superconvergence for state and adjoint $L^2$ error, especially with improved post-processing;
• Results consistent for both Dirichlet and Neumann boundary conditions, with the latter requiring a modified projection relation to maintain constraint satisfaction.

5. Comparative Advantages and Extensibility

The GDM framework offers several strategic and methodological advantages:

• Unified analysis, extensibility, robustness: Once the four GDM properties are established for a numerical scheme, the full body of GDM theoretical results applies. This makes the framework particularly amenable to the analysis of novel, hybrid, or locally refined schemes, as well as methods on general nonstructured or polytopal meshes.
• Wide-ranging applicability: GDM encompasses control, constraint, nonlinear, and non-smooth problems, in both stationary and time-dependent settings.
• Error estimates provide explicit dependence on mesh, solution regularity, and scheme properties: This enables practical a priori error control, facilitating adaptive strategies.

The extension to more complex or coupled systems, as well as non-standard (e.g., multi-physics) and mixed-dimensional settings, is direct in principle given appropriate operator definitions that embed the scheme into the GD structure.

6. Research Impact and Further Developments

GDM has had significant influence, as evidenced by its application to:

• Superconvergence for post-processed controls in non-conforming and hybrid schemes (first results of this kind);
• Robust error analysis for complex variational inequalities and free boundary problems, such as nonlinear Signorini and seepage models;
• Nonlinear, possibly degenerate or singular diffusion models via general operators and nontraditional meshes.

Papers by Droniou, Eymard, Herbin, and collaborators have provided both the foundational framework (see [DEH15], “The Gradient Discretisation Method”) and its practical and theoretical extensions. The methodology is now a reference point for unified convergence studies and code verification across a spectrum of modern PDE simulation tasks.

7. Key Formulas and Indicators

The most critical GDM indicators and formulas include:

• PDE error estimate:

$$\| \Pi_\mathcal{D} \psi_\mathcal{D} - \psi \| + \| \nabla_\mathcal{D} \psi_\mathcal{D} - \nabla \psi \| \lesssim S_\mathcal{D}(\psi) + W_\mathcal{D}(A\nabla \psi)$$

• Superconvergence for post-processed control:

$$\| \widetilde{u} - \widetilde{u}_h \|_{L^2} \leq C h^{2-\frac{1}{2^*}}$$

(typically $O(h^2)$ for quasi-uniform meshes)

• Definitions of $S_\mathcal{D}$ and $W_\mathcal{D}$:
  • $S_\mathcal{D}(\psi)$: interpolant/consistency error,
  • $W_\mathcal{D}(\zeta)$: limit-conformity error,
  • with both scaling as $O(h)$ for standard grids/schemes.

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The Gradient Discretisation Method stands as a central and versatile analytical and computational paradigm, allowing systematic, rigorous, and practically robust handling of a very broad family of numerical discretisations for linear and nonlinear PDEs, control, and constraint problems, including those posing significant analytical challenges such as mesh nonconformity, heterogeneity, or strong nonlinearity. Its adoption enables unified theoretical guarantees across applications and a sound basis for implementing and verifying advanced numerical procedures.