Polyhedral Spline Finite Elements

TL;DR

This paper introduces multivariate spline finite elements constructed from mollified low-smoothness splines and B-splines, offering high smoothness and strong approximation power suitable for isogeometric analysis on general partitions.

Contribution

It presents a novel construction of multivariate spline finite elements with high smoothness and approximation capabilities for general polyhedral partitions.

Findings

Splines are constructed by mollifying low-smoothness splines with B-splines.

The resulting splines have high smoothness and strong approximation power.

Evaluation is efficient, making them suitable for collocation and adaptive methods.

Abstract

Spline functions have long been used in numerically solving differential equations. Recently it revives as isogeometric analysis, which uses NURBS for both parametrization and element functions. In this paper, we introduce some multivariate spline finite elements on general partitions. These splines are constructed by mollifying splines of low smooth order with B-splines. They have both high smooth order and strong approximation power. The low smooth order splines can be chosen flexibly, for example polynomials on individual cells of general polyhedral partitions. Evaluation of such splines is not difficult, so the obtained spline elements are suitable for the collocation method and adaptive computation.