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 nonlinear, steadystate or timedependent. The schemes may be conforming or nonconforming, and may rely on very general polygonal or polyhedral meshes (or may even be meshless).
Differential equations  

Navier–Stokes differential equations used to simulate airflow around an obstruction.  
Classification  
Types


Relation to processes 

Solution  
General topics 

Solution methods 

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 nonlinear. For linear problems, stationary or transient, error estimates can be established based on three indicators specific to the GDM [1] (the quantities , and , see below). For nonlinear problems, the proofs are based on compactness techniques and do not require any nonphysical strong regularity assumption on the solution or the model data.[2] Nonlinear models for which such convergence proof of the GDM have been carried out comprise: the Stefan problem which is modelling a melting material, twophase flows in porous media, the Richards equation of underground water flow, the fully nonlinear 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 MultiPoint 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 [4] to this model is:
In a nutshell, the GDM for such a model consists in selecting a finitedimensional 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" (vectorvalued 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,[5] 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 GDconsistency and the limitconformity 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 (6)) remains bounded.
GDconsistency
For all , (defined by (7)).
Limitconformity
For all , (defined by (8)). 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).
Piecewise constant reconstruction (needed for some nonlinear problems)
Let be a gradient discretisation as defined above. The operator is a piecewise constant reconstruction if there exists a basis of and a family of disjoint subsets of such that for all , where is the characteristic function of .
Some nonlinear problems with complete convergence proofs of the GDM
We review some problems for which the GDM can be proved to converge when the above core properties are satisfied.
Nonlinear stationary diffusion problems
In this case, the GDM converges under the coercivity, GDconsistency, limitconformity and compactness properties.
pLaplace problem for p > 1
In this case, the core properties must be written, replacing by , by and by with , and the GDM converges only under the coercivity, GDconsistency and limitconformity properties.
Linear and nonlinear heat equation
In this case, the GDM converges under the coercivity, GDconsistency (adapted to spacetime problems), limitconformity and compactness (for the nonlinear case) properties.
Degenerate parabolic problems
Assume that and are nondecreasing Lipschitz continuous functions:
Note that, for this problem, the piecewise constant reconstruction property is needed, in addition to the coercivity, GDconsistency (adapted to spacetime problems), limitconformity and compactness properties.
Review of some numerical methods which are GDM
All the methods below satisfy the first four core properties of GDM (coercivity, GDconsistency, limitconformity, compactness), and in some cases the fifth one (piecewise constant reconstruction).
Galerkin methods and conforming finite element methods
Let be spanned by the finite basis . The Galerkin method in is identical to the GDM where one defines
In this case, is the constant involved in the continuous Poincaré inequality, and, for all , (defined by (8)). Then (4) and (5) are implied by Céa's lemma.
The "masslumped" finite element case enters the framework of the GDM, replacing by , where is a dual cell centred on the vertex indexed by . Using mass lumping allows to get the piecewise constant reconstruction property.
Nonconforming finite element
On a mesh which is a conforming set of simplices of , the nonconforming finite elements are defined by the basis of the functions which are affine in any , 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 NavierStokes equations). Then the method enters the GDM framework with the same definition as in the case of the Galerkin method, except for the fact that must be understood as the "broken gradient" of , in the sense that it is the piecewise constant function equal in each simplex to the gradient of the affine function in the simplex.
Mixed finite element
The mixed finite element method consists in defining two discrete spaces, one for the approximation of and another one for .[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.
Discontinuous Galerkin method
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.
Mimetic finite difference method and nodal mimetic finite difference method
This family of methods is introduced by [Brezzi et al][9] and completed in [Lipnikov et al].[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].[2]
See also
References
 R. Eymard, C. Guichard, and R. Herbin. Smallstencil 3d schemes for diffusive flows in porous media. M2AN, 46:265–290, 2012.
 J. Droniou, R. Eymard, T. Gallouët, and R. Herbin. Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. (M3AS), 23(13):2395–2432, 2013.
 J. Leray and J. Lions. Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de MintyBrowder. Bull. Soc. Math. France, 93:97–107, 1965.
 H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
 G. Strang. Variational crimes in the finite element method. In The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972), pages 689–710. Academic Press, New York, 1972.
 M. Crouzeix and P.A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7(R3):33–75, 1973.
 P.A. Raviart and J. M. Thomas. A mixed finite element method for 2nd order elliptic problems. In Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pages 292–315. Lecture Notes in Math., Vol. 606. Springer, Berlin, 1977.
 D. A. Di Pietro and A. Ern. Mathematical aspects of discontinuous Galerkin methods, volume 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg, 2012.
 F. Brezzi, K. Lipnikov, and M. Shashkov. Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal., 43(5):1872–1896, 2005.
 K. Lipnikov, G. Manzini, and M. Shashkov. Mimetic finite difference method. J. Comput. Phys., 257Part B:1163–1227, 2014.
External links
 The Gradient Discretisation Method by Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard and Raphaèle Herbin