三角化曲面上的多元样条的Bernstein约束复形

中文摘要 AI 辅助

我们引入了Bernstein约束复形，这是一种有限元微分复形的表示，其中三角剖分的每个三角形保留其自身的Bernstein-Bézier系数，全局连续性仅通过附着在边和顶点上的光滑性泛函来施加。标量样条的连续性、样条向量场的切向连续性、样条向量场的法向连续性以及更高阶的光滑性仅在边泛函上有所不同。不构造全局基。结构结果是系数层面的交换恒等式：在每条边上，Bernstein导数的光滑性泛函是其参数的光滑性泛函的显式组合，通过单变量Bernstein差分矩阵实现。这给出了一个局部证明，即光滑性矩阵的核构成一个子复形，对于de Rham迹和逐分量$C^r$样条轮廓，以及一个可计算的候选光滑轮廓的行空间测试。我们以这种形式实现了最低阶Powell-Sabin正合序列，并通过秩确认其正合性。约束Galerkin问题在具有精确光滑性方程的破碎系数中求解；我们证明了与协调方法的等价性，通过伪逆接口系统处理半定单元算子，并给出了一个奇异束，其有限谱是约束Maxwell谱。在协变和逆变曲面Piola映射下，de Rham序列的参考泛函不变地转移到协调弯曲三角剖分上。实验比较了零空间方法、Awanou、Lai和Wenston的增广拉格朗日迭代以及鞍点系统的直接求解，在相同问题上进行，并涵盖平面源和特征值问题、$C^1$双调和求解、精确球面和嵌入双曲抛物面。

英文摘要

We introduce Bernstein constraint complexes, a representation of finite element differential complexes in which every triangle of a triangulation keeps its own Bernstein--Bézier coefficients and global continuity is imposed only through smoothness functionals attached to edges and vertices. Continuity of scalar splines, tangential continuity of spline vector fields, normal continuity of spline vector fields, and higher smoothness differ only in the edge functionals. No global basis is constructed. The structural result is a coefficient-level commuting identity: on every edge the smoothness functionals of a Bernstein derivative are explicit combinations of the smoothness functionals of its argument, through univariate Bernstein difference matrices. This gives a local proof that the kernels of the smoothness matrices form a subcomplex, for the de Rham traces and for componentwise $C^r$ spline profiles, and a computable row-space test for candidate smooth profiles. We realize the lowest-order Powell--Sabin exact sequence in this form and confirm its exactness by ranks. The constrained Galerkin problem is solved in the broken coefficients with exact smoothness equations; we prove equivalence with the conforming method, treat semidefinite element operators through a pseudoinverse interface system, and give a singular pencil whose finite spectrum is the constrained Maxwell spectrum. Under the covariant and contravariant surface Piola maps the reference functionals of the de Rham sequence transfer unchanged to conforming curved triangulations. Experiments compare the null-space method, the augmented Lagrangian iteration of Awanou, Lai and Wenston, and direct solution of the saddle system on the same problems, and cover planar source and eigenvalue problems, a $C^1$ biharmonic solve, the exact sphere, and an embedded hyperboloid.

发表机构

• Texas A&M University（德克萨斯农工大学）

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