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
CD
SD
WD
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
\Omega\subsetRd
where
f\inL2(\Omega)
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
D=(XD,0,\PiD,\nablaD)
XD,0
\PiD~:~XD,0\toL2(\Omega)
XD,0
\Omega
\nablaD~:~XD,0\toL2(\Omega)d
XD,0
\Omega
\Vert\nablaD ⋅
\Vert | |
L2(\Omega)d |
XD,0
The related Gradient Scheme for the approximation of (2) is given by: find
u\inXD,0
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
\nablaDu
\PiDu
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 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
p
Let
(Dm)m\inN
The sequence
(C | |
Dm |
)m\inN
For all
\varphi\in
1 | |
H | |
0(\Omega) |
\limm\toinfty
S | |
Dm |
(\varphi)=0
For all
\varphi\inH\operatorname{div}(\Omega)
\limm\toinfty
W | |
Dm |
(\varphi)=0
For all sequence
(um)m\inN
um\in
X | |
Dm,0 |
m\inN
(\Vertum
\Vert | |
Dm |
)m\inN
(\Pi | |
Dm |
um)m\inN
L2(\Omega)
Let
D=(XD,0,\PiD,\nablaD)
\PiD
(ei)i\in
XD,0
(\Omegai)i\in
\Omega
\chi | |
\Omegai |
\Omegai
We review some problems for which the GDM can be proved to converge when the above core properties are satisfied.
-\operatorname{div}(Λ(\overline{u})\nabla\overline{u})=f
In this case, the GDM converges under the coercivity, GD-consistency, limit-conformity and compactness properties.
-\operatorname{div}\left(|\nabla\overline{u}|p-2\nabla\overline{u}\right)=f
In this case, the core properties must be written, replacing
L2(\Omega)
Lp(\Omega)
1 | |
H | |
0(\Omega) |
1,p | |
W | |
0(\Omega) |
H\operatorname{div}(\Omega)
p' | |
W | |
\operatorname{div} |
(\Omega)
\partialt\overline{u}-\operatorname{div}(Λ(\overline{u})\nabla\overline{u})=f
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.
Assume that
\beta
\zeta
\partialt\beta(\overline{u})-\Delta\zeta(\overline{u})=f
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).
Let
Vh\subset
1 | |
H | |
0(\Omega) |
(\psii)i\in
Vh
XD,0=\{u=(ui)i\in\}=RI,
\PiDu=\sumi\inui\psii
\nablaDu=\sumi\inui\nabla\psii.
In this case,
CD
\varphi\inH\operatorname{div}(\Omega)
WD(\varphi)=0
The "mass-lumped"
P1
\PiDu
\Omegai
i\inI
On a mesh
T
Rd
P1
(\psii)i\in
K\inT
\nabla\psii
\psii
The mixed finite element method consists in defining two discrete spaces, one for the approximation of
\nabla\overline{u}
\overline{u}
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.
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''].