Gradient discretisation method

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 then be known to converge on these problems; this occurs in the case of the conforming Finite Elements, the Raviart--Thomas Mixed Finite Elements, or the $\Pfe_1$ 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

Consider a simple Poisson problem in a domain $\Omega \subset \mathbb {R} ^{d}$, with homogeneous Dirichlet boundary conditionspartial differential equation

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

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 $\nabla _{D}~:~X_{D,0}\to L^{2}(\Omega )^{d}\$isalinearmappingwhichreconstructs,fromanelementof<math>X_{D,0}$, 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 :

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

and

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

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 forall $\varphi \in H_{\rm {div}}(\Omega )$, $W_{D}(\varphi )$ tends to 0.

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