Gradient discretisation method: Difference between revisions

Revision as of 01:30, 15 November 2016

Revision as of 01:35, 15 November 2016

Line 45:

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

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

W_{D}W_D(\varphi) = \max_{v\in X_{D,0}\setminus\{0\}}\frac{

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

which measures the defect of conformity.

Line 59:

The sequence <math>(C_{D_m})_{m\in\mathbb{N}}</math> (defined by (6)) remains bounded.

=== GD-consistency ===

For all <math>\varphi\in H^1_0(\Omega)</math>, <math>\lim_{m\to\infty} S_{D_m} (\varphi) = 0</math> (defined by (7)).

=== Limit-conformity ===

For all <math>\varphi\in H_\operatorname{\rm div}(\Omega)</math>, <math>\lim_{m\to\infty} W_{D_m}(\varphi) = 0</math> (defined by (8)).

=== Compactness (needed for some nonlinear problems)===

For all sequence <math>(u_m)_{m\in\mathbb{N}}</math> such that <math>u_m \in X_{D_m,0} </math> for all <math>m\in\mathbb{N}</math> and <math>(\Vert u_m \Vert_{D_m})_{m\in\mathbb{N}}</math> is bounded, then the sequence <math>(\Pi_{D_m} u_m)_{m\in\mathbb{N}}</math> is relatively compact in <math>L^2(\Omega)</math> (this property implies the coercivity property).

=== Piecewise constant reconstruction (needed for some nonlinear problems)===

Let <math>D = (X_{D,0}, \Pi_D,\nabla_D)</math> be a gradient discretisation as defined above.

The operator <math>\Pi_D</math> is a piecewise constant reconstruction if there exists a basis <math>(e_i)_{i\in B}</math> of <math>X_{D,0}</math> and a family of disjoint subsets <math>(\Omega_i)_{i\in B}</math> of <math>\Omega</math> such that <math>\Pi_D u = \sum_{i\in B}u_i\chi_{\Omega_i}</math> for all <math>u=\sum_{i\in B} u_i e_i\in X_{D,0}</math>, where <math>\chi_{\Omega_i}</math> is the characteristic function of <math>\Omega_i</math>.

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

Line 77:

=== Nonlinear stationary diffusion problems ===

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

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

=== ''p''-Laplace problem for ''p'' <math>p> 1</math>===

:<math>\quad \qquad \qquad -{\rm operatorname{div}(|\nabla \overline{u}|^{p-2}\nabla \overline{u}) = f</math>

In this case, the core properties must be written, replacing <math>L^2(\Omega)</math> by <math>L^p(\Omega)</math>, <math>H^1_0(\Omega)</math> by <math>W^{1,p}_0(\Omega)</math> and <math>H_\operatorname{\rm div}(\Omega)</math> by <math>W_\operatorname{\rm div}^{p'}(\Omega)</math> with <math>\frac 1 {p} +\frac 1 {p'}=1</math>, and the GDM converges only under the coercivity, GD-consistency and limit-conformity properties.

=== Linear and nonlinear heat equation ===

:<math>\quad \qquad \qquad \partial_t \overline{u}-{\rm \operatorname{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.

Line 109:

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>,

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

*<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 inequality, and, for all <math>\varphi\in H_H_operatorname{\rm div}(\Omega)</math>, <math>W_{D}(\varphi) = 0</math> (defined by (8)).

The "mass-lumped" <math>P^1</math> finite element case enters the framework of the GDM, replacing <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.

Line 126:

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

=== Mimetic Finitefinite Differencedifference method and nodal mimetic Mimeticfinite Finite Differencedifference method ===

==See also==