Weitere Informationen zum ausgewählten Kolloquiumsvortrag:

The benefits of smoothness in Isogeometric Analysis

Dr. Espen Sande

Monday, 2021-01-18 11:00

Splines are piecewise polynomial functions that are glued together with a given smoothness. When using them in a numerical method, the availability of proper error estimates is of utmost importance. Classical error estimates for spline approximation are expressed in terms of

(a) a certain power of the maximal grid spacing,

(b) an appropriate derivative of the function to be approximated, and

(c) a "constant" which is independent of the previous quantities but usually depends on the degree and smoothness of the spline.

An explicit expression of the constant in (c) is rarely available in the literature, because it is a minor issue in standard approximation analysis. There they are mainly interested in the approximation power of spline spaces of a fixed degree. However, one of the most interesting features in the emerging field of Isogeometric Analysis is krefinement, which denotes degree elevation with increasing interelement smoothness. The above mentioned error estimates are not sufficient to explain the benefits of approximation under k-refinement so long as it is not well understood how the constant in (c) behaves. In this talk we provide error estimates for k-refinement on arbitrary grids with an explicit constant that is, in certain cases, sharp. These estimates are in fact good enough to cover convergence to eigenfunctions of classical differential operators. This forms a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature. Seite 2 Moreover, we discuss how these error estimates can be used to mathematically justify the benefits of spline approximation under k-refinement. Specifically, by comparing the constant for spline approximation of maximal smoothness with a lower bound on the constant for continuous and discontinuous spline approximation, we show that k-refinement provides better approximation per degree of freedom in almost all cases of practical interest. This talk is based on work performed in collaboration with Andrea Bressan and Carla Manni and Hendrik Speleers.