Gradient discretisation method

In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent discretisation schemes for diffusion problems of different 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. Owing to these core properties, it is possible to prove the convergence of a GDM for standard elliptic and parabolic problems, linear or non-linear. Then any scheme entering the GDM framework is known to converge on these problems; this occurs in the case of the conforming Finite Elements, the Raviart—Thomas Mixed Finite Elements, or the $P^{1}$ non-conforming Finite Elements, or in the case of more recent schemes, such as the Hybrid Mixed Mimetic or Nodal Mimetic methods, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes.

The example of a linear diffusion problem

Let us consider Poisson's equation in a bounded open domain $\Omega \subset \mathbb {R} ^{d}$, with homogeneous Dirichlet boundary condition

$\quad (1)\qquad \qquad -\Delta {\overline {u}}=f,$

where $f\in L^{2}(\Omega )$, and the solution ${\overline {u}}\in H_{0}^{1}(\Omega )$ is such that

$\quad (2)\qquad \qquad \forall {\overline {v}}\in H_{0}^{1}(\Omega ),\qquad \int _{\Omega }\nabla {\overline {u}}(x)\cdot \nabla {\overline {v}}(x)dx=\int _{\Omega }f(x){\overline {v}}(x)dx.$

The GDM is defined by $D=(X_{D,0},\Pi _{D},\nabla _{D})$, where:

• the set of discrete unknowns $X_{D,0}$ is a finite dimensional real vector space,
• the function reconstruction $\Pi _{D}~:~X_{D,0}\to L^{2}(\Omega )$ is a linear mapping that reconstructs, from an element of $X_{D,0}$, a function over $\Omega$,
• the gradient reconstruction $\nabla _{D}~:~X_{D,0}\to L^{2}(\Omega )^{d}$ is a linear mapping which reconstructs, from an element of $X_{D,0}$, a "gradient" (vector-valued function) over $\Omega$. This gradient reconstruction must be chosen such that $\Vert \cdot \Vert _{D}:=\Vert \nabla _{D}\cdot \Vert _{L^{2}(\Omega )^{d}}$ is a norm on $X_{D,0}$.

The Gradient Scheme for the approximation of (2) is given by: find $u\in X_{D,0}$ such that

$\quad (3)\qquad \qquad \forall v\in X_{D,0},\qquad \int _{\Omega }\nabla _{D}u(x)\cdot \nabla _{D}v(x)dx=\int _{\Omega }f(x)\Pi _{D}v(x)dx.$

Then there holds the following error estimate, inspired by [Strang,1972]

$\quad (4)\qquad \qquad \Vert \nabla {\overline {u}}-\nabla _{D}u_{D}\Vert _{L^{2}(\Omega )^{d}}\leq W_{D}(\nabla {\overline {u}})+2S_{D}({\overline {u}}),$

and

$\quad (5)\qquad \qquad \Vert {\overline {u}}-\Pi _{D}u_{D}\Vert _{L^{2}(\Omega )}\leq C_{D}W_{D}(\nabla {\overline {u}})+(C_{D}+1)S_{D}({\overline {u}}),$

defining:

$\quad (6)\qquad \qquad C_{D}=\max _{v\in X_{D,0}\setminus \{0\}}{\frac {\Vert \Pi _{D}v\Vert _{L^{2}(\Omega )}}{\Vert v\Vert _{D}}},$

$\quad (7)\qquad \qquad \forall \varphi \in H_{0}^{1}(\Omega ),\,S_{D}(\varphi )=\min _{v\in X_{D,0}}\left(\Vert \Pi _{D}v-\varphi \Vert _{L^{2}(\Omega )}+\Vert \nabla _{D}v-\nabla \varphi \Vert _{L^{2}(\Omega )^{d}}\right),$

$\quad (8)\qquad \qquad \forall \varphi \in H_{\rm {div}}(\Omega ),\,W_{D}(\varphi )=\sup _{u\in X_{D,0}\setminus \{0\}}{\frac {\left|\int _{\Omega }\left(\nabla _{D}u(x)\cdot \varphi (x)+\Pi _{D}u(x){\rm {div}}\varphi (x)\right)dx\right|}{\Vert u\Vert _{D}}}$

Then the core properties which are sufficient for the convergence of the method are, for a family of GDM, that $C_{D}$ remains bounded, that, for all $\varphi \in H_{0}^{1}(\Omega )$, $S_{D}(\varphi )$ tends to 0, and that for all $\varphi \in H_{\rm {div}}(\Omega )$, $W_{D}(\varphi )$ tends to 0. These core properties are not sufficient for proving the convergence of the GDM when applied to some nonlinear problems (nonlinear diffusion, degenerate parabolic problems...), and we detail in the next section the set of core properties which are required for covering a larger range of problems.

The core properties allowing for the convergence of a GDM

These properties are defined for a family $(D_{m})_{m\in \mathbb {N} }$ of GDM defined as above (generally associated with a sequence of regular meshes whose size tends to 0).

Coercivity

The sequence $(C_{D_{m}})_{m\in \mathbb {N} }$ (defined by (6)) remains bounded.

GD-consistency

For all $\varphi \in H_{0}^{1}(\Omega )$, $\lim _{m\to \infty }S_{D_{m}}(\varphi )=0$ (defined by (7)).

Limit-conformity

For all $\varphi \in H_{\rm {div}}(\Omega )$, $\lim _{m\to \infty }W_{D_{m}}(\varphi )=0$ (defined by (8)).

Compactness (needed for some nonlinear problems)

For all sequence $(u_{m})_{m\in \mathbb {N} }$ such that $u_{m}\in X_{D_{m},0}$ for all $m\in \mathbb {N}$ and $(\Vert u_{m}\Vert _{D_{m}})_{m\in \mathbb {N} }$ is bounded, then the sequence $(\Pi _{D_{m}}u_{m})_{m\in \mathbb {N} }$ is relatively compact in $L^{2}(\Omega )$.

Piecewise constant reconstruction (needed for some nonlinear problems)

Let $D=(X_{D,0},\Pi _{D},\nabla _{D})$ be a gradient discretisation as defined above. The operator $\Pi _{D}$ is a piecewise constant reconstruction if there exists a basis $(e_{i})_{i\in B}$ of $X_{D,0}$ and a family of disjoint subsets $(\Omega _{i})_{i\in B}$ of $\Omega$ such that $\Pi _{D}u=\sum _{i\in B}u_{i}\chi _{\Omega _{i}}$ for all $u=\sum _{i\in B}u_{i}e_{i}\in X_{D,0}$, where $\chi _{\Omega _{i}}$ is the characteristic function of $\Omega _{i}$.

Review of some problems which may be approximated by a GDM

We pass into review a few problems for which the GDM can be proved to converge when the above core properties are satisfied.

Nonlinear stationnary diffusion problems

$\quad \qquad \qquad -{\rm {div}}(\Lambda ({\overline {u}})\nabla {\overline {u}})=f$

In this case, the GDM converges under the coercivity, GD-consistency, limit-conformity and compactness properties.

$p$-Laplace problem for $p>1$

$\quad \qquad \qquad -{\rm {div}}(|\nabla {\overline {u}}|^{p-2}\nabla {\overline {u}})=f$

In this case, the core properties must be written, replacing $L^{2}(\Omega )$ by $L^{p}(\Omega )$, $H_{0}^{1}(\Omega )$ by $W_{0}^{1,p}(\Omega )$ and $H_{\rm {div}}(\Omega )$ by $W_{\rm {div}}^{p'}(\Omega )$ with ${\frac {1}{p}}+{\frac {1}{p'}}=1$, and the GDM converges only under the coercivity, GD-consistency and limit-conformity properties.

Linear and nonlinear heat equation

$\quad \qquad \qquad \partial _{t}{\overline {u}}-{\rm {div}}(\Lambda ({\overline {u}})\nabla {\overline {u}})=f$

In this case, the GDM converges under the coercivity, GD-consistency, limit-conformity and compactness (for the nonlinear case) properties.

Degenerate parabolic problems

Assume that $\beta$ and $\zeta$ are nondecreasing Lipschitz continuous functions:

$\quad \qquad \qquad \partial _{t}\beta ({\overline {u}})-\Delta \zeta ({\overline {u}})=f$

Note that, for this problem, the piecewise constant reconstruction property is needed, in addition to the coercivity, GD-consistency, 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 (coercivity, GD-consistency, limit-conformity, compactness), and in some cases piecewise constant reconstruction.

Galerkin methods and conforming Finite Element methods

Let $V_{h}\subset H_{0}^{1}(\Omega )$ be spanned by the finite basis $(\psi _{i})_{i\in I}$. The Galerkin method in $V_{h}$ is identical to the GDM where one defines

• $X_{D,0}=\{u=(u_{i})_{i\in I}\}=\mathbb {R} ^{I}$,
• $\Pi _{D}u=\sum _{i\in I}u_{i}\psi _{i}$
• $\nabla _{D}u=\sum _{i\in I}u_{i}\nabla \psi _{i}$.

In this case, $C_{D}$ is the constant involved in the continuous Poincaré's inequality, and, for all $\varphi \in H_{\rm {div}}(\Omega )$, $W_{D}(\varphi )=0$ (defined by (8)).

The "mass-lumped" $P^{1}$ finite element case enters in the framework of the GDM: it suffices to replace $\Pi _{D}u$ by ${\widetilde {\Pi }}_{D}u=\sum _{i\in I}u_{i}\chi _{\Omega _{i}}$, where $\Omega _{i}$ is a dual cell centred on the vertex indexed by $i\in I$. Using mass lumping allows to get the piecewise constant reconstruction property.

Nonconforming $P^{1}$ finite element

On a mesh $T$ which is a conforming set of simplices of $\mathbb {R} ^{d}$, the nonconforming $P^{1}$ finite elements are defined by the basis $(\psi _{i})_{i\in I}$ of the functions which are affine in any $K\in T$, 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 into the GDM framework with the same definition as in the case of the Galerkin method, except the fact that $\nabla \psi _{i}$ must be understood as the "broken gradient" of $\psi _{i}$, 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 $\nabla {\overline {u}}$ and another one for ${\overline {u}}$. It suffices to use the discrete relations between these approximations for defining a GDM. Using the low degree Raviart-Thomas mixed finite elements allows to get the piecewise constant reconstruction property.

See also

• Finite element method

References

• Strang, G.. (1972) "Variational crimes in the finite element method" The mathematical foundations of the finite element method with applications to partial differential equations, p. 689–710.

External links

• The Gradient Discretisation Method by Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard and Raphaèle Herbin