# Graduate Seminar "Aktuelle Themen der Numerik"

### Thursday, May 08, 2014, 02:00 pm

Tree Approximation for hp-Adaptivity

Prof. Peter Binev (University of South Carolina)

**Abstract:**

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 (2*N*−1)σ* _{n } */(

*N*-

*n*+1),

where σ

*,*

_{n }*n*≤

*N*, is the best possible approximation of complexity

*n*.

**Time**: 02:00 pm

**Location**: Room 149, Hauptgebäude, RWTH Aachen, Templergraben 55, 52056 Aachen