Approximating Gradients with Continuous Piecewise Polynomial Functions
Approximating Gradients with Continuous Piecewise Polynomial Functions
复制标题
用连续分段多项式函数逼近梯度
DOI：
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发表时间：
2014
影响因子：
3
通讯作者：
A. Veeser
中科院分区：
文献类型：
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作者：
A. Veeser
受二阶椭圆问题的有限元方法的启发，我们通过简单网格上的连续分段多项式函数来分析目标函数的梯度逼近。主要结果是，全局最佳近似误差相当于元素上局部最佳近似误差的适当总和。因此，要求连续性不会降低局部逼近能力，并且不连续分段多项式本质上不提供额外的逼近能力，即使对于固定网格也是如此。该结果意味着在整个允许的平滑度范围内分段正则性方面存在误差界限。此外，它允许在梯度的自适应树近似中使用简单的局部误差函数。
Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is that the global best approximation error is equivalent to an appropriate sum in terms of the local best approximation errors on elements. Thus, requiring continuity does not downgrade local approximation capability and discontinuous piecewise polynomials essentially do not offer additional approximation power, even for a fixed mesh. This result implies error bounds in terms of piecewise regularity over the whole admissible smoothness range. Moreover, it allows for simple local error functionals in adaptive tree approximation of gradients.