Gradient discretisation method: Difference between revisions

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:<math>\quad (9) \qquad \qquad \begin{align} &&\frac 1 2 [S_D(\overline{u}) + W_D(\nabla \overline{u})] \\ &\le & \Vert \overline{u} - \Pi_D u_D\Vert_{L^2(\Omega)} + \Vert \nabla \overline{u} - \nabla_D u_D\Vert_{L^2(\Omega)^d} \\ &\le &(C_D+3) [S_D(\overline{u}) + W_D(\nabla \overline{u})].\end{align} </math>

Then the core properties which are necessary and 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. TheseMore generally, these three core properties are sufficient to prove the convergence of the GDM for linear problems and for some nonlinear problems like the $p$-Laplace problem. For nonlinear problems (such as nonlinear diffusion, degenerate parabolic problems...), we add in the next section two other core properties which may be required.

==The core properties allowing for the convergence of a GDM==