Sobolev Regularity of Isogeometric Finite Element Spaces with Degenerate Geometry Map

TL;DR

This paper studies the Sobolev regularity of isogeometric finite element functions with degenerate geometry maps, focusing on the necessary conditions for second derivatives to ensure accurate approximation of elliptic PDEs.

Contribution

It extends the understanding of regularity conditions for isogeometric spaces with degenerate geometry maps, especially for higher-order PDEs.

Findings

Tightened C1-conditions for D-patches to ensure second derivative integrability.

Identified the impact of degenerate points on Sobolev regularity.

Provided guidelines for finite element approximation of biharmonic equations with degenerate geometries.

Abstract

We investigate Sobolev regularity of bivariate functions obtained in Isogeometric Analysis when using geometry maps that are degenerate in the sense that the first partial derivatives vanish at isolated points. In particular, we show how the known C1-conditions for D-patches have to be tightened to guarantee square integrability of second partial derivatives, as required when computing finite element approximations of elliptic fourth order PDEs like the biharmonic equation.