Gradient discretisation method: Difference between revisions

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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. ThenAs a consequence, any scheme entering the GDM framework is then known to converge on these problems; this occurs in the case of the conforming Finite Elements, the Raviart—Thomas Mixed Finite Elements, or the <math>P^1</math> 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==

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:<math>\quad (2) \qquad \qquad \forall \overline{v} \in H^1_0(\Omega),\qquad \int_{\Omega} \nabla \overline{u}(x)\cdot\nabla \overline{v}(x) dx = \int_{\Omega} f(x)\overline{v}(x) dx. </math>

TheA GDMGradient Discretization (GD) is defined by a triplet <math>D = (X_{D,0},\Pi_D,\nabla_D)</math>, where:

* the set of discrete unknowns <math>X_{D,0}</math> is a finite dimensional real vector space,

* the function reconstruction <math>\Pi_D~:~X_{D,0}\to L^2(\Omega)</math> is a linear mapping that reconstructs, from an element of <math>X_{D,0}</math>, a function over <math>\Omega</math>,

* the gradient reconstruction <math>\nabla_D~:~X_{D,0}\to L^2(\Omega)^d</math> is a linear mapping which reconstructs, from an element of <math>X_{D,0}</math>, a "gradient" (vector-valued function) over <math>\Omega</math>. This gradient reconstruction must be chosen such that <math>\Vert \cdot \Vert_{D} := \Vert \nabla_D \cdot \Vert_{L^2(\Omega)^d}</math> is a norm on <math>X_{D,0}</math>.

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

:<math>\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. </math>

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

:<math>\quad (4) \qquad \qquad \Vert \nabla \overline{u} - \nabla_D u_D\Vert_{L^2(\Omega)^d}

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defining:

:<math>\quad (6) \qquad \qquad C_D = \max_{v\in X_{D,0}\setminus\{0\}}\frac {\Vert \Pi_D v\Vert_{L^2(\Omega)}} {\Vert \nabla_D v \Vert_{DL^2(\Omega)^d}}, </math>

which measures the coercivity (discrete Poincaré constant),

:<math>\quad (7) \qquad \qquad

\forall \varphi\in H^1_0(\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), </math>

which measures the interpolation error,

:<math>\quad (8) \qquad \qquad

\forall \varphi\in H_{\rm div}(\Omega),\,

W_{D}(\varphi) = \sup_max_{uv\in X_{D,0}\setminus\{0\}}\frac{

\left|\int_\Omega \left(\nabla_D uv(x)\cdot\varphi(x) + \Pi_D uv(x) {\rm div}\varphi(x)\right) dx \right|}{\Vert \nabla_D uv \Vert_{DL^2(\Omega)^d}}, </math>

which measures the defect of conformity.

Then the core properties which are sufficient for the convergence of the method are, for a family of GDMGDs, thatthe <math>C_D</math> remains boundedcoercivity, that,the forGD-consistency alland <math>\varphi\inthe H^1_0(\Omega)</math>,limit-conformity <math>S_{D}(\varphi)</math> tends to 0properties, andas thatdefined for all <math>\varphi\in H_{\rmthe div}(\Omega)</math>,next <math>W_{D}(\varphi)</math> tends to 0section. These three core properties are not sufficient forto provingprove the convergence of the GDM whenfor appliedlinear toproblems. someFor nonlinear problems (nonlinear diffusion, degenerate parabolic problems...), we add in the next section two other core properties which may be required.

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 familyLet <math>(D_m)_{m\in\mathbb{N}}</math> be a family of GDMGDs, defined as above (generally associated with a sequence of regular meshes whose size tends to 0).

=== Coercivity ===

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==Review of some problems which may be approximated by a GDM==

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

=== Nonlinear stationnarystationary diffusion problems ===

:<math>\quad \qquad \qquad -{\rm div}(\Lambda(\overline{u})\nabla \overline{u}) = f</math>

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:<math>\quad \qquad \qquad \partial_t \overline{u}-{\rm div}(\Lambda(\overline{u})\nabla \overline{u}) = f</math>

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.

=== Degenerate parabolic problems ===

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:<math>\quad \qquad \qquad \partial_t \beta(\overline{u})-\Delta \zeta(\overline{u}) = f</math>

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.

==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 the fifth one (piecewise constant reconstruction).

===[[Galerkin methodsmethod]]s and conforming Finite Element methods===

Let <math>V_h\subset H^1_0(\Omega)</math> be spanned by the finite basis <math>(\psi_i)_{i\in I}</math>. The [[Galerkin method]] in <math>V_h</math> is identical to the GDM where one defines

*<math>X_{D,0} = \{ u = (u_i)_{i\in I} \} = \mathbb{R}^I</math>,

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*<math>\nabla_D u = \sum_{i\in I} u_i \nabla\psi_i</math>.

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

The "mass-lumped" <math>P^1</math> finite element case enters in the framework of the GDM: it suffices to, replacereplacing <math>\Pi_D u</math> by <math>\widetilde{\Pi}_D u = \sum_{i\in I} u_i \chi_{\Omega_i}</math>, where <math>\Omega_i</math> is a dual cell centred on the vertex indexed by <math>i\in I</math>. Using mass lumping allows to get the piecewise constant reconstruction property.

=== Nonconforming <math>P^1</math> finite element ===

On a mesh <math>T</math> which is a conforming set of simplices of <math>\mathbb{R}^d</math>, the nonconforming <math>P^1</math> finite elements are defined by the basis <math>(\psi_i)_{i\in I}</math> of the functions which are affine in any <math>K\in T</math>, 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 for the fact that <math>\nabla\psi_i</math> must be understood as the "broken gradient" of <math>\psi_i</math>, 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 <math>\nabla \overline{u}</math> and another one for <math>\overline{u}</math>.

It suffices to use the discrete relations between these approximations forto definingdefine a GDM. Using the low degree Raviart-Thomas mixed finite elements allows to get the piecewise constant reconstruction property.

=== Mimetic Finite Difference method and nodal Mimetic Finite Difference method ===