Jump-preserving polynomial interpolation in non-manifold polyhedra

Abstract

We construct a piecewise-polynomial interpolant $u\mapsto \hat{I}u$ for functions $u:\hat{I}\text{c◯}\setminus \hat{I}“\to \mathbb{R}$, where $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ is a Lipschitz polyhedron and $\hat{I}“\subset \hat{I}\text{c◯}$ 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, ${\hat{I}}^2=\hat{I}$, and $\hat{I}$ preserves homogeneous boundary values and jumps of its argument on $\hat{I}“$. As an application, we obtain a bounded discrete right-inverse of the "jump" operator across $\hat{I}“$, and an error estimate for a Galerkin scheme to solve a second-order elliptic PDE in $\hat{I}\text{c◯}$ with a prescribed jump across $\hat{I}“$.