An adaptive approach using HB-spline discretizations

ANNALISA BUFFA; EDUARDO M. GARAU

Trondheim

Congreso; III International Conference on Isogemetric Analysis; 2015

International Center for Numerical Methods in Engineering (CIMNE)

Local adaptivity in numerical methods for partial differential equations makes possible to solve real problems leading to a suitable approximation of the desired solution without exceeding the limits of available software. When considering isogemetric methods, from a theoretical point of view, the design of efficient and robust strategies for local refinement constitutes a challenging problem because the tensor product structure of splines is broken. In this work, we consider hierarchical B-spline (HB-spline) spaces (Vuong et al., 2011 and Giannelli et al., 2012) in order to discretize partial differential equations. We analize the local approximation properties of the discrete spaces through the study of quasi-interpolant operators. Furthermore, we will remark that some HB-splines can be removed from the hierarchical basis without losing ability of approximation. Moreover, this new space, that we study and characterize, seems to be suited for adaptivity. Additionally, in this context, we discuss both theoretical and practical aspects of an adaptive loop of the form SOLVE -> ESTIMATE -> MARK -> REFINE. In particular, a posteriori error estimations and strategies for refinement in this framework are interesting and useful points to analyze. The first theoretical results on these topics are given in (Buffa and Giannelli, 2015), where simple residual element based error estimators are used.