Approximation error estimates and inverse inequalities for B-splines of maximum smoothness

TL;DR

This paper establishes degree-independent approximation error estimates and inverse inequalities for maximum smoothness B-splines, facilitating efficient isogeometric analysis and PDE discretizations.

Contribution

It introduces new approximation error estimates and inverse inequalities for B-splines that are independent of polynomial degree, applicable to maximum smoothness splines.

Findings

Error estimates do not depend on polynomial degree

Inverse inequalities hold in a subspace of B-spline space

Results enable faster iterative methods for PDEs

Abstract

In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard Sobolev norms and semi-norms. The presented approximation error estimates do not depend on the polynomial degree of the splines but only on the grid size. We will see that the approximation lives in a subspace of the classical B-spline space. We show that for this subspace, there is an inverse inequality which is also independent of the polynomial degree. As the approximation error estimate and the inverse inequality show complementary behavior, the results shown in this paper can be used to construct fast iterative methods for solving problems arising from isogeometric discretizations of partial differential equations.