论文标题
稳定的本地通勤投影仪和最佳$ hp $ $近似值估算$ {\ boldsymbol h}（\ mathrm {curl}）$
A stable local commuting projector and optimal $hp$ approximation estimates in ${\boldsymbol H}(\mathrm{curl})$
论文作者
论文摘要
我们从无限维sobolev space $ {\ boldsymbol h}（\ mathrm {curl}）$设计一个操作员到其由nédélec零工综合的有限尺寸子空间，该子空间在四面体网上在四面体网络上具有以下属性： H}（\ Mathrm {curl}）$，包括在边界的一部分上施加的边界条件； 2）在每个网格元素的邻域中定义它； 3）它基于简单的分段多项式投影； 4）它在$ {\ boldsymbol l}^2 $ -Norm中是稳定的，直至数据振荡； 5）它具有最佳（本地最佳）近似特性； 6）它可以在$ {\ boldsymbol h}（\ mathrm {div}）$上与其同胞运算符的通勤属性满足属性； 7）它是一个投影仪，即，它留下了已在nédélec分段多项式空间中的完整对象。该操作员可用于与$ {\ boldsymbol H}（\ Mathrm {curl}）$ Space相关的数值分析的各个部分。我们在这里特别采用它来建立以下两个结果：i）等于全球最佳，切向 - 跟踪和卷曲，局部最佳，最佳，不受约束的近似值，$ {\ boldsymbol h}（\ mathrm {curl}）$，包括数据振荡项； ii）完全$ h $ - 和$ p $ - （网状大小和多项式 - 基准 - ）最佳近似范围仅在最小的sobolev规则下有效，仅请求元素。由于独立兴趣，我们还证明了$ {\ boldsymbol h}（\ boldsymbol h}（\ mathrm {curl}）$的单个四面体上的卷发约束和不受限制的最佳评估的等效性，包括$ HP $数据示波项。
We design an operator from the infinite-dimensional Sobolev space ${\boldsymbol H}(\mathrm{curl})$ to its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: 1) it is defined over the entire ${\boldsymbol H}(\mathrm{curl})$, including boundary conditions imposed on a part of the boundary; 2) it is defined locally in a neighborhood of each mesh element; 3) it is based on simple piecewise polynomial projections; 4) it is stable in the ${\boldsymbol L}^2$-norm, up to data oscillation; 5) it has optimal (local-best) approximation properties; 6) it satisfies the commuting property with its sibling operator on ${\boldsymbol H}(\mathrm{div})$; 7) it is a projector, i.e., it leaves intact objects that are already in the Nédélec piecewise polynomial space. This operator can be used in various parts of numerical analysis related to the ${\boldsymbol H}(\mathrm{curl})$ space. We in particular employ it here to establish the two following results: i) equivalence of global-best, tangential-trace-and curl-constrained, and local-best, unconstrained approximations in ${\boldsymbol H}(\mathrm{curl})$ including data oscillation terms; and ii) fully $h$- and $p$- (mesh-size- and polynomial-degree-) optimal approximation bounds valid under the minimal Sobolev regularity only requested elementwise. As a result of independent interest, we also prove a $p$-robust equivalence of curl-constrained and unconstrained best-approximations on a single tetrahedron in the ${\boldsymbol H}(\mathrm{curl})$-setting, including $hp$ data oscillation terms.
