In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, and may rely on very general polygonal or polyhedral meshes (or may even be meshless).
Some core properties are required to prove the convergence of a GDM. These core properties enable complete proofs of convergence of the GDM for elliptic and parabolic problems, linear or non-linear. For linear problems, stationary or transient, error estimates can be established based on three indicators specific to the GDM (the quantities , and , see below). For non-linear problems, the proofs are based on compactness techniques and do not require any non-physical strong regularity assumption on the solution or the model data. Non-linear models for which such convergence proof of the GDM have been carried out comprise: the Stefan problem which is modelling a melting material, two-phase flows in porous media, the Richards equation of underground water flow, the fully non-linear Leray—Lions equations.
Any scheme entering the GDM framework is then known to converge on all these problems. This applies in particular to conforming Finite Elements, Mixed Finite Elements, nonconforming Finite Elements, and, in the case of more recent schemes, the Discontinuous Galerkin method, Hybrid Mixed Mimetic method, the Nodal Mimetic Finite Difference method, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes
The example of a linear diffusion problem
Consider Poisson's equation in a bounded open domain , with homogeneous Dirichlet boundary condition
where . The usual sense of weak solution to this model is:
In a nutshell, the GDM for such a model consists in selecting a finite-dimensional space and two reconstruction operators (one for the functions, one for the gradients) and to substitute these discrete elements in lieu of the continuous elements in (2). More precisely, the GDM starts by defining a Gradient Discretization (GD), which is a triplet , where:
the set of discrete unknowns is a finite dimensional real vector space, the function reconstruction is a linear mapping that reconstructs, from an element of , a function over , the gradient reconstruction is a linear mapping which reconstructs, from an element of , a "gradient" (vector-valued function) over . This gradient reconstruction must be chosen such that is a norm on .
The related Gradient Scheme for the approximation of (2) is given by: find such that
The GDM is then in this case a nonconforming method for the approximation of (2), which includes the nonconforming finite element method. Note that the reciprocal is not true, in the sense that the GDM framework includes methods such that the function cannot be computed from the function .
The following error estimate, inspired by G. Strang's second lemma, holds
and
defining:
which measures the coercivity (discrete Poincaré constant),
which measures the interpolation error,
which measures the defect of conformity.
Note that the following upper and lower bounds of the approximation error can be derived:
Then the core properties which are necessary and sufficient for the convergence of the method are, for a family of GDs, the coercivity, the GD-consistency and the limit-conformity properties, as defined in the next section. More generally, these three core properties are sufficient to prove the convergence of the GDM for linear problems and for some nonlinear problems like the -Laplace problem. For nonlinear problems such as nonlinear diffusion, degenerate parabolic problems..., we add in the next section two other core properties which may be required.
The core properties allowing for the convergence of a GDM
Let be a family of GDs, defined as above (generally associated with a sequence of regular meshes whose size tends to 0).
Coercivity
The sequence (defined by ()) remains bounded.
GD-consistency
For all , (defined by ()).
Limit-conformity
For all , (defined by ()). This property implies the coercivity property.
Compactness (needed for some nonlinear problems) For all sequence such that for all and is bounded, then the sequence is relatively compact in (this property implies the coercivity property).