Volume 12, Issue 1
The Instability in the Dimensions of Spline Spaces over $T$-Meshes with Nested $T$-Cycles

Chongjun Li & Pengxiao Wang

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 187-211.

Published online: 2018-09

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  • Abstract

This paper studies on the dimensions of spline spaces over some given $T$-meshes. Using the smoothing cofactor-conformality method, we study the instability in the dimensions of the spline spaces over $T$-meshes with 2-nested and 3-nested $T$-cycles. We define a singularity factor of each simple $T$-cycle, the instability and the structure's degeneration are associated with the singularity factors. In order to get a stable dimension formula over $T$-mesh with a $N$-nested $T$-cycle, a constraint on the $T$-mesh is introduced. Finally, a possible degeneration for a case of parallel $T$-cycles is illustrated.

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@Article{NMTMA-12-187, author = {}, title = {The Instability in the Dimensions of Spline Spaces over $T$-Meshes with Nested $T$-Cycles}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {12}, number = {1}, pages = {187--211}, abstract = {

This paper studies on the dimensions of spline spaces over some given $T$-meshes. Using the smoothing cofactor-conformality method, we study the instability in the dimensions of the spline spaces over $T$-meshes with 2-nested and 3-nested $T$-cycles. We define a singularity factor of each simple $T$-cycle, the instability and the structure's degeneration are associated with the singularity factors. In order to get a stable dimension formula over $T$-mesh with a $N$-nested $T$-cycle, a constraint on the $T$-mesh is introduced. Finally, a possible degeneration for a case of parallel $T$-cycles is illustrated.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2017-0110}, url = {http://global-sci.org/intro/article_detail/nmtma/12697.html} }
TY - JOUR T1 - The Instability in the Dimensions of Spline Spaces over $T$-Meshes with Nested $T$-Cycles JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 187 EP - 211 PY - 2018 DA - 2018/09 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2017-0110 UR - https://global-sci.org/intro/article_detail/nmtma/12697.html KW - AB -

This paper studies on the dimensions of spline spaces over some given $T$-meshes. Using the smoothing cofactor-conformality method, we study the instability in the dimensions of the spline spaces over $T$-meshes with 2-nested and 3-nested $T$-cycles. We define a singularity factor of each simple $T$-cycle, the instability and the structure's degeneration are associated with the singularity factors. In order to get a stable dimension formula over $T$-mesh with a $N$-nested $T$-cycle, a constraint on the $T$-mesh is introduced. Finally, a possible degeneration for a case of parallel $T$-cycles is illustrated.

Chongjun Li & Pengxiao Wang. (2020). The Instability in the Dimensions of Spline Spaces over $T$-Meshes with Nested $T$-Cycles. Numerical Mathematics: Theory, Methods and Applications. 12 (1). 187-211. doi:10.4208/nmtma.OA-2017-0110
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