Dorfler marking is cardinality-optimal up to mesh closure

Let ${\mathcal{T}}^{+}=\mathcal{T}\oplus {\mathcal{T}}_{\ast }$ and let $U,U_{\ast },U^{+}$ be the exact Galerkin solutions. Best approximation on the overlay gives $e_{{\mathcal{T}}^{+}}\le e_{{\mathcal{T}}_{\ast }}\le \lambda e_{\mathcal{T}}$. By [F2], $\mathrm{\parallel }\mathrm{\nabla }(U^{+}-U){\mathrm{\parallel }}_2^2=e_{\mathcal{T}}^2-e_{{\mathcal{T}}^{+}}^2\ge (1-{\lambda }^2)e_{\mathcal{T}}^2\mathrm{.}$ By [F3], this correction square is at most $C_{\mathrm{l}\mathrm{o}\mathrm{c}}{\eta }_{\mathcal{T}}(\mathcal{R}{)}^2$, where $\mathcal{R}$ is the set of coarse cells subdivided in ${\mathcal{T}}^{+}$. By [F1], ${\eta }_{\mathcal{T}}^2\le C_{\mathrm{e}\mathrm{f}\mathrm{f}}{e}_{\mathcal{T}}^2$. Combining proves ${\eta }_{\mathcal{T}}(\mathcal{R}{)}^2\ge {\theta }_{\ast }{\eta }_{\mathcal{T}}^2$. If $e_{\mathcal{T}}=0$, then the comparison condition and all indicators are zero and the empty set is Dörfler.

The bulk algorithm sorts nonnegative weights in decreasing order. Among all sets of a fixed size $m$, the first $m$ sorted weights have maximal sum: replacing any omitted larger weight for a selected smaller one cannot decrease the sum, and repeating yields the prefix. Therefore the shortest prefix reaching threshold ${\theta }_E{\eta }_{\mathcal{T}}^2$ has minimum cardinality among all Dörfler sets. Since $\mathcal{R}$ is such a set whenever ${\theta }_E\le {\theta }_{\ast }$, $\mathrm{\mid }{M}_E\mathrm{\mid }\le \mathrm{\mid }\mathcal{R}\mathrm{\mid }$. The overlay cardinality bound [F4] gives $\mathrm{\mid }\mathcal{R}\mathrm{\mid }\le \mathrm{\#}{\mathcal{T}}^{+}-\mathrm{\#}\mathcal{T}\le \mathrm{\#}{\mathcal{T}}_{\ast }-\mathrm{\#}{\mathcal{T}}_0$.