Gradient discretisation method
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 [1] (the quantities
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.
[2] 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.
[3] 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 Consider Poisson's equation in a bounded open domain
The usual sense of weak solution [4] 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
Note that the reciprocal is not true, in the sense that the GDM framework includes methods such that the function
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.
be a family of GDs, defined as above (generally associated with a sequence of regular meshes whose size tends to 0).
We review some problems for which the GDM can be proved to converge when the above core properties are satisfied.
In this case, the GDM converges under the coercivity, GD-consistency, limit-conformity and compactness properties.
, and the GDM converges only under the coercivity, GD-consistency and limit-conformity properties.
In this case, the GDM converges under the coercivity, GD-consistency (adapted to space-time problems), limit-conformity and compactness (for the nonlinear case) properties.
are nondecreasing Lipschitz continuous functions: Note that, for this problem, the piecewise constant reconstruction property is needed, in addition to the coercivity, GD-consistency (adapted to space-time problems), limit-conformity and compactness properties.
All the methods below satisfy the first four core properties of GDM (coercivity, GD-consistency, limit-conformity, compactness), and in some cases the fifth one (piecewise constant reconstruction).
finite element case enters the framework of the GDM, replacing
Using mass lumping allows to get the piecewise constant reconstruction property.
, and whose value at the centre of gravity of one given face of the mesh is 1 and 0 at all the others (these finite elements are used in [Crouzeix et al][6] for the approximation of the Stokes and Navier-Stokes equations).
The mixed finite element method consists in defining two discrete spaces, one for the approximation of
[7] It suffices to use the discrete relations between these approximations to define a GDM.
Using the low degree Raviart–Thomas basis functions allows to get the piecewise constant reconstruction property.
The Discontinuous Galerkin method consists in approximating problems by a piecewise polynomial function, without requirements on the jumps from an element to the other.
[8] It is plugged in the GDM framework by including in the discrete gradient a jump term, acting as the regularization of the gradient in the distribution sense.
[10] It allows the approximation of elliptic problems using a large class of polyhedral meshes.
The proof that it enters the GDM framework is done in [Droniou et al].
of the p -Laplace problem on the domain [0,1] with (black line) and approximate one (blue line) computed with the first degree discontinuous Galerkin method plugged into the GDM (uniform mesh with 6 elements).
