Instance Optimality of the Adaptive Maximum Strategy

Abstract

In this paper, we prove that the standard adaptive finite element method with a (modified) maximum marking strategy is instance optimal for the total error, being the square root of the squared energy error plus the squared oscillation. This result will be derived in the model setting of Poisson’s equation on a polygon, linear finite elements, and conforming triangulations created by newest vertex bisection.

Appendix: A Slightly Modified Marking

In this section, we propose a routine MARK, resorting to slightly modified accumulated indicators, that can be implemented in ${\mathcal {O}}(\#\mathcal {P}_k)$ operations. The important fact is that Proposition 5.1 remains valid, which ensures the instance optimality in Sect. 7 also for this modified marking step.

To this end, we first compute a modified maximal accumulated indicator $\bar{{E}}_{\mathcal {P}_k}^2$. This value can be determined with the help of the following recursive routine.

Algorithm 7.1

(Maximal Indicator) Set $\bar{{E}}_{\mathcal {P}_k}^2:=0$ and call $\mathsf max-ind (P,0)$ for all $P\in (\mathcal {P}_k^{++}\setminus \mathcal {P}_k)$ with no parents in $\mathcal {P}_k^{++}\setminus \mathcal {P}_k$.

In general, we have $\bar{{E}}_{\mathcal {P}_k}^2\ne \bar{\mathcal {E}}_{\mathcal {P}_k}^2$, since $C\in \mathcal {P}_k^{++}\setminus \mathcal {P}_k$ may have two parents $P,P' \in \mathcal {P}_k^{++}\setminus \mathcal {P}_k$. However, such a $C$ cannot have children in $\mathcal {P}^{++}\setminus \mathcal {P}$ as is illustrated in Fig. 8, and so we conclude that

$$\begin{aligned} \bar{{E}}_{\mathcal {P}_k}^2\ge \frac{1}{2}\bar{\mathcal {E}}_{\mathcal {P}_k}^2. \end{aligned}$$

Fig. 8

${\mathcal {T}}(\mathcal {P})$ (solid lines), $P,P',Q \in \mathcal {P}^{++}\setminus \mathcal {P}=\mathsf{\mathrm {midpt} (\mathcal {S}({\mathcal {T}}(\mathcal {P})))}$ with $\{P,P'\} = \mathtt{parent}(Q)$, and $\{\Box \}=\mathtt{child}(Q).$

Next, the sets ${\mathcal {M}}_{k}$ and $\widetilde{\mathcal {M}}_{k}$ are determined by running the following routine.

Algorithm 7.2

(Marking) Set ${\mathcal {M}}_{k}:=\widetilde{\mathcal {M}}_{k}:=\emptyset$ and call $\mathsf accum-est (P,0)$ for all $P \in \mathcal {P}_k^{++}\setminus \mathcal {P}_k$ with no parents in $\mathcal {P}_k^{++}\setminus \mathcal {P}_k$.

One verifies that ${\mathcal {M}}_{k}, \widetilde{\mathcal {M}}_{k}\subset \mathcal {P}_k^{++}\setminus \mathcal {P}_k$, $\mathcal {P}_k \oplus {\mathcal {M}}_{k}= \mathcal {P}_k \cup \widetilde{\mathcal {M}}_{k}$, and

$$\begin{aligned} {\mathcal {E}}_{\mathcal {P}_k}^2 \big ( (\mathcal {P}_k \oplus {\mathcal {M}}_{k})\setminus \mathcal {P}_k\big ) \ge \mu \,\# {\mathcal {M}}_{k}\, \bar{{E}}_{\mathcal {P}_k}^2\ge \frac{1}{2}\mu \,\# {\mathcal {M}}_{k}\, \bar{\mathcal {E}}_{\mathcal {P}_k}^2; \end{aligned}$$

i.e. Proposition 5.1 is still valid.

Finally, the work needed for this evaluation of MARK scales linearly with $\# \mathcal {P}_k$. Indeed, the number of times that a $P \in \mathcal {P}_k^{++}\setminus \mathcal {P}_k$ is accessed by the flow of computation is proportional to the number of calls $\text {accum-est}(P,\cdot )$ (being one), plus the number of its children in $\mathcal {P}_k^{++}\setminus \mathcal {P}_k$ (being uniformly bounded).