Session B6 - Multiresolution and Adaptivity in Numerical PDEs
July 14, 17:30 ~ 17:55
Near-Best Approximation on Adaptive Partitions
University of South Carolina, United States - email@example.com
We consider adaptive approximation of a function on a given domain $\Omega$ using piecewise polynomial functions. To study this approximation, the partitioning of $\Omega$ is related to building a binary tree $T$ whose leaves represent the elements of the partition. The problem of approximating a function is presented then as the process of identifying a tree $T$ and the related approximating polynomials $P(\Delta)$ for each of its leaves $\Delta$. We consider two cases of adaptivity: the h-adaptivity in which the polynomials $P(\Delta)$ are of the same order and the hp-adaptivity in which the orders of $P(\Delta)$ may vary for different $\Delta$-s. We use error functionals to drive the adaptive process and prove that the presented algorithms provide a near-best approximation in both cases.