Gradient discretisation method: Difference between revisions

Revision as of 10:10, 15 November 2016

Revision as of 10:11, 15 November 2016

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

which measures the defect of conformity.

Note that (4) and (5) are implied by [[Céa's lemma]] in the particular case of a conforming approximation (in the sense that the operators <math>\Pi_D</math> and <math>\nabla_D</math> are such that, for all <math>v\in X_{D,0}</math>, <math>\Pi_D v \in H^1_0(\Omega)</math> and <math>\nabla_D v = \nabla(\Pi_D v)</math>; then <math>C_D</math> is replaced by the Poincaré constant and <math>W_D(\cdot)= 0</math>, leading to a Galerkin method as described below).

Then the core properties which are 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. These three core properties are sufficient to prove the convergence of the GDM for linear problems. For nonlinear problems (nonlinear diffusion, degenerate parabolic problems...), we add in the next section two other core properties which may be required.