Gradient discretisation method
In numerical mathematics, the gradient discretisation method 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.
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 . 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 conditionwhere. 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 and to substitute these discrete elements in lieu of the continuous elements in. More precisely, the GDM starts by defining a Gradient Discretization, 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" over. This gradient reconstruction must be chosen such that is a norm on.
The GDM is then in this case a nonconforming method for the approximation of, 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,
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.Coercivity
The sequence remains bounded.GD-consistency
For all, .Limit-conformity
For all, .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 .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 non-linear 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, GD-consistency, limit-conformity and compactness properties.''p''-Laplace 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, GD-consistency and limit-conformity properties.Linear and nonlinear heat equation
In this case, the GDM converges under the coercivity, GD-consistency, limit-conformity and compactness 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, GD-consistency, limit-conformity and compactness properties.
Review of some numerical methods which are GDM
All the methods below satisfy the first four core properties of GDM, and in some cases the fifth one.[Galerkin method]s and conforming finite element methods
Let be spanned by the finite basis . The Galerkin method in is identical to the GDM where one definesIn this case, is the constant involved in the continuous Poincaré inequality, and, for all, . Then and are implied by Céa's lemma.
The "mass-lumped" 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 . 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.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.