Tree Approximation for hp - Adaptivity IMI P REPRINT S ERIES Peter Binev C OLLEGE OF A RTS AND S CIENCES U NIVERSITY OF S OUTH C AROLINA

The hp-adaptive approximation is formulated as an approximation problem on a full binary tree T , where for each of the leaves ∆ an order p(∆) ≥ 1 is assigned in such a way that the sum of all such p(∆) does not exceed N , called complexity of the approximation. The leaves ∆ correspond to the cells of the partition, while p(∆) is the order of the polynomial used for the local approximation on ∆. Devising an incremental algorithm for near-best adaptive approximation for the problem of finding the best possible tree T and assignments p(∆) leads to building a construction that attaches a ghost tree with p(∆) leaves to each leaf ∆ of T with p(∆) > 1. The resulting full binary tree T that 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 our approximation of complexity N is bounded by 2N−1 N−n+1σn, where σn, n ≤ N , is the best possible approximation of complexity n.