We design an operator from the infinite-dimensional Sobolev space ${\boldsymbol H}(\mathrm{curl})$ to its finite-dimensional subspace formed by the N\'ed\'elec 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 N\'ed\'elec 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.

翻译：我们设计了一个算子，将无限维 Sobolev 空间 ${\boldsymbol H}(\mathrm{curl})$ 映射到由四面体网格上的 Nédélec 分片多项式构成的有限维子空间，该算子具有以下性质：1) 定义在整个 ${\boldsymbol H}(\mathrm{curl})$ 空间上，包括施加在部分边界上的边界条件；2) 在每个网格单元邻域内局部定义；3) 基于简单的分片多项式投影；4) 在 ${\boldsymbol L}^2$ 范数下稳定（至多允许数据振荡项）；5) 具有最优（局部最优）逼近性质；6) 满足与其在 ${\boldsymbol H}(\mathrm{div})$ 中的兄弟算子的换位性质；7) 是一个投影算子，即保持已在 Nédélec 分片多项式空间中的对象不变。该算子可应用于与 ${\boldsymbol H}(\mathrm{curl})$ 空间相关的数值分析多个领域。我们特别利用它建立以下两个结果：i) 在 ${\boldsymbol H}(\mathrm{curl})$ 中，全局最优（受切向迹与旋度约束）与局部最优（无约束）逼近的等价性（包含数据振荡项）；ii) 在仅要求单元级最小 Sobolev 正则性条件下，完全 $h$ 和 $p$（网格尺寸和多项式次数）最优逼近界限的成立性。作为独立意义的结果，我们还在 ${\boldsymbol H}(\mathrm{curl})$ 框架下证明了单个四面体上旋度约束最优逼近与无约束最优逼近的 $p$ 鲁棒等价性（包含 $hp$ 数据振荡项）。