有界矩形区域上非均匀十字交叉三角剖分的二次样条拟插值误差分析
Error analysis for quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of bounded rectangular domains
AI总结：
针对矩形域上的非均匀十字交叉剖分，分析了两类C1二次样条拟插值算子（基于B样条与离散泛函线性组合）的逼近误差，证明了其无穷范数小且与剖分无关，对二次多项式精确，从而对光滑函数达到最优逼近阶，并给出了依赖于光滑度和网格比的最优误差界。
摘要：
给定矩形域Ω上的非均匀十字交叉剖分，我们分析了定义在Ω上的函数f与两类C1二次样条拟插值算子（QIs）之间的误差，这些QIs是B样条与离散泛函作为系数的线性组合。主要新颖之处在于B样条的支撑集包含在Ω内，且数据点也位于Ω内部或边界上。此外，这些QIs的无穷范数很小且不依赖于三角剖分：由于两个QIs在二次多项式上精确，它们对光滑函数给出了最优逼近阶。我们的分析针对f及其一阶和二阶偏导数进行，并特别努力给出了依赖于f光滑性和三角剖分网格比的最佳可能误差界。
英文摘要：
Given a non-uniform criss-cross partition of a rectangular domain $Ω$, we analyse the error between a function $f$ defined on $Ω$ and two types of $C^1$-quadratic spline quasi-interpolants (QIs) obtained as linear combinations of B-splines with discrete functionals as coefficients. The main novelties are the facts that supports of B-splines are contained in $Ω$ and that data sites also lie inside or on the boundary of $Ω$. Moreover, the infinity norms of these QIs are small and do not depend on the triangulation: as the two QIs are exact on quadratic polynomials, they give the optimal approximation order for smooth functions. Our analysis is done for $f$ and its partial derivatives of the first and second orders and a particular effort has been made in order to give the best possible error bounds in terms of the smoothness of $f$ and of the mesh ratios of the triangulation.
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