We construct a piecewise-polynomial interpolant $u \mapsto \Pi u$ for functions $u:\Omega \setminus \Gamma \to \mathbb{R}$, where $\Omega \subset \mathbb{R}^d$ is a Lipschitz polyhedron and $\Gamma \subset \Omega$ is a possibly non-manifold $(d-1)$-dimensional hypersurface. This interpolant enjoys approximation properties in relevant Sobolev norms, as well as a set of additional algebraic properties, namely, $\Pi^2 = \Pi$, and $\Pi$ preserves homogeneous boundary values and jumps of its argument on $\Gamma$. As an application, we obtain a bounded discrete right-inverse of the "jump" operator across $\Gamma$, and an error estimate for a Galerkin scheme to solve a second-order elliptic PDE in $\Omega$ with a prescribed jump across $\Gamma$.

翻译：我们构造了一个分片多项式插值算子 $u \mapsto \Pi u$，用于定义在 $\Omega \setminus \Gamma$ 上的函数 $u:\Omega \setminus \Gamma \to \mathbb{R}$，其中 $\Omega \subset \mathbb{R}^d$ 是一个Lipschitz多面体，$\Gamma \subset \Omega$ 是一个可能非流形的 $(d-1)$ 维超曲面。该插值算子在相关Sobolev范数下具有逼近性质，并满足一组额外的代数性质，即 $\Pi^2 = \Pi$，且 $\Pi$ 保持其自变量在 $\Gamma$ 上的齐次边界值和跳跃。作为应用，我们获得了跨越 $\Gamma$ 的“跳跃”算子的有界离散右逆，以及一个Galerkin格式的误差估计，该格式用于求解 $\Omega$ 中具有指定跨 $\Gamma$ 跳跃的二阶椭圆偏微分方程。