三维空间多项式次稳健势和通量重建的稳定破缺$H^1$和$\bf H(\mathrm{div})$多项式延拓
Stable broken $H^1$ and $\bf H(\mathrm{div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions
摘要：
我们研究扩展的分段多项式数据规定的面孔，并可能在一个补丁的单形共享一个顶点的元素。在$H^1$设置中，我们寻找其跨面跳跃是规定的函数，而在${\bf H}(\mathrm{div})$设置中，法向分量跳跃和分段发散是规定的。我们表明稳定的意义上说，分段多项式空间的相同程度的数据的极小值是从属于在整个破碎的$H^1$和${\bf H}(\mathrm{div})$空间的极小值在破碎的能量范数。我们的证明是建设性的，并产生独立于多项式次数的常数。这些结果的一个特定的应用是在后验误差分析，其中目前的结果证明多项式的程度，强大的潜在和通量重建的效率。
英文摘要：
We study extensions of piecewise polynomial data prescribed on faces and possibly in elements of a patch of simplices sharing a vertex. In the $H^1$ setting, we look for functions whose jumps across the faces are prescribed, whereas in the ${\bf H}(\mathrm{div})$ setting, the normal component jumps and the piecewise divergence are prescribed. We show stability in the sense that the minimizers over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken $H^1$ and ${\bf H}(\mathrm{div})$ spaces. Our proofs are constructive and yield constants independent of the polynomial degree. One particular application of these results is in a posteriori error analysis, where the present results justify polynomial-degree-robust efficiency of potential and flux reconstructions.
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