Jump-preserving polynomial interpolation in non-manifold polyhedra

TL;DR

This paper develops a polynomial interpolation method for functions with jumps across non-manifold hypersurfaces in polyhedral domains, enabling accurate approximation and solving PDEs with discontinuities.

Contribution

It introduces a jump-preserving polynomial interpolant for non-manifold hypersurfaces, with proven approximation properties and algebraic features, facilitating PDE solutions with discontinuities.

Findings

Constructed a piecewise-polynomial interpolant with approximation guarantees.

Established algebraic properties including idempotency and boundary/jump preservation.

Provided error estimates for PDE schemes with jump conditions.

Abstract

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 $\Gamma$.