Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes
For researchers in isogeometric analysis, this provides theoretical bounds on linear dependence in locally refined spline spaces, which is crucial for ensuring stable numerical methods.
The paper determines the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (6) and Locally Refinable B-splines (8) on LR-meshes, showing that MS B-splines have a significantly higher risk of linear dependency.
The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeoemtric analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. In this paper we address the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (MS B-splines) and for Locally Refinable B-splines (LR B-splines) on LR-meshes. We show that the minimal number is six for MS B-splines, and eight for LR B-splines. The risk of linear dependency is consequently significantly higher for MS B-splines than for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent.