4,190 research outputs found
On the dimension of spline spaces on planar T-meshes
We analyze the space of bivariate functions that are piecewise polynomial of
bi-degree \textless{}= (m, m') and of smoothness r along the interior edges of
a planar T-mesh. We give new combinatorial lower and upper bounds for the
dimension of this space by exploiting homological techniques. We relate this
dimension to the weight of the maximal interior segments of the T-mesh, defined
for an ordering of these maximal interior segments. We show that the lower and
upper bounds coincide, for high enough degrees or for hierarchical T-meshes
which are enough regular. We give a rule of subdivision to construct
hierarchical T-meshes for which these lower and upper bounds coincide. Finally,
we illustrate these results by analyzing spline spaces of small degrees and
smoothness
Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes
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
Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering
There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering
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