Session B6 - Multiresolution and Adaptivity in Numerical PDEs
July 15, 14:30 ~ 14:55 - Room T1
Adaptive Hierarchical B-Splines: Convergence and Optimality
Ricardo H Nochetto
University of Maryland, College Park, USA - firstname.lastname@example.org
We present and analyze an adaptive hierarchical B-splines method, of any (fixed) order and maximal regularity, for linear elliptic equations. We first set the framework for approximation with hierarchical B-splines, and next develop an a posteriori error estimator based on solutions of discrete local problems on stars. We next propose a novel refinement strategy to increase the local resolution of the spaces and derive a contraction property for D\"orfler's marking which implies linear convergence. We also study the complexity of our refinement strategy in terms of marked cells, as well as the optimal relation of energy error vs degrees of freedom for a suitable nonlinear class of functions.
Joint work with Pedro Morin (Universidad Nacional del Litoral, Argentina) and M. Sebastian Pauletti (Universidad Nacional del Litoral, Argentina).