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Volume 21, Issue 3
Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures

Meng Wu, Bernard Mourrain, André Galligo & Boniface Nkonga

DOI: 10.4208/cicp.OA-2016-0030

Commun. Comput. Phys., 21 (2017), pp. 835-866.

Published online: 2018-04

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  • Abstract

Motivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and $C^r$ parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the $L^2$-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

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@Article{CiCP-21-835, author = {Wu , MengMourrain , BernardGalligo , André and Nkonga , Boniface}, title = {Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures}, journal = {Communications in Computational Physics}, year = {2018}, volume = {21}, number = {3}, pages = {835--866}, abstract = {

Motivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and $C^r$ parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the $L^2$-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0030}, url = {http://global-sci.org/intro/article_detail/cicp/11262.html} }
TY - JOUR T1 - Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures AU - Wu , Meng AU - Mourrain , Bernard AU - Galligo , André AU - Nkonga , Boniface JO - Communications in Computational Physics VL - 3 SP - 835 EP - 866 PY - 2018 DA - 2018/04 SN - 21 DO - http://doi.org/10.4208/cicp.OA-2016-0030 UR - https://global-sci.org/intro/article_detail/cicp/11262.html KW - AB -

Motivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and $C^r$ parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the $L^2$-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

Wu , MengMourrain , BernardGalligo , André and Nkonga , Boniface. (2018). Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures. Communications in Computational Physics. 21 (3). 835-866. doi:10.4208/cicp.OA-2016-0030
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