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The following error estimate, inspired by G. Strang's second lemma,<ref>'''G. Strang.''' Variational crimes in the finite element method.'' In The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972)'', pages 689–710. Academic Press, New York, 1972.</ref> holds
:<math>\quad (4) \qquad \qquad W_D(\nabla \overline{u}) \le \Vert \nabla \overline{u} - \nabla_D u_D\Vert_{L^2(\Omega)^d}
\le W_D(\nabla \overline{u}) + 2 S_D(\overline{u}), </math>
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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 it is easy to deduce the following upper and lower bounds of the approximation error:
:<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 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.
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