Tree Approximation for hp-Adaptivity
Peter Binev · SIAM Journal on Numerical Analysis · 2018
The hp-adaptive approximation is formulated as an approximation problem on a full binary tree $T$, where for each of the leaves $\Delta$ an order $p(\Delta)\ge1$ is assigned in such a way that the sum of all orders $p(\Delta)$ does not exceed $N$, which is called the complexity of the approximation. The leaves $\Delta$ correspond to the cells of the partition, while $p(\Delta)$ is the dimension of the polynomial space used for the local approximation on $\Delta$. Devising an incremental algorithm for near-best adaptive approximation for the problem of finding the best possible tree $T$ and assignments $p(\Delta)$ leads to building a construction that attaches a ghost tree with $p(\Delta)$ leaves to each leaf $\Delta$ of $T$ with $p(\Delta)>1$. The resulting full binary tree $\mathcal{T}$ has at most $N$ leaves and can be used as a proxy of $T$ for assembling hp-adaptive procedures. Under the standard assumptions about the local errors, we prove that the error of our approximation of complexity $N$ is bounded by $\frac{2N-1}{N-n+1}\sigma_n$, where $\sigma_n$, $n\le N$, is the error of the best possible approximation of complexity $n$.