Jump-preserving polynomial interpolation in non-manifold polyhedra

TL;DR

The paper tackles the challenge of discretizing PDEs on domains with a non-manifold crack or screen $\Gamma$ by constructing a jump-preserving, piecewise-polynomial interpolant $\Pi_h$ mapping $H^1(\Omega\setminus\Gamma)$ into a finite element space $V^p(\Omega_h;\Gamma)$. It introduces a robust, jump-aware interpolation framework that is idempotent ($$Π_h^2=Π_h$$), preserves homogeneous boundary data and the jumps across $\Gamma$, and provides both local and global Sobolev stability/approximation estimates in $H^t$ and $H^s$ spaces. A key methodological advance is the extension of Scott--Zhang style interpolation to non-manifold hypersurfaces via a multi-sides/bridges construction, including primal/dual bases and bridge functions, to manage multiple approach directions at a node. The framework yields a bounded discrete right-inverse to the jump operator and enables Galerkin error analysis for elliptic PDEs with prescribed jumps on $\Gamma$, with convergence $\|u-u_h\|_{H^1(\Omega\setminus\Gamma)} \lesssim h^{l-1}|u|_{H^l(\Omega\setminus\Gamma)}$ for $1\le l\le p+1$. Overall, the results significantly advance stable discretization techniques for non-manifold fracture-like geometries and thin obstacles in boundary-element and finite-element contexts.

Abstract

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

Figures (5)

• Figure 1: A non-manifold surface $\Gamma \subset \mathbb{R}^3$ and its conforming triangular mesh.
• Figure 2: A two-dimensional situation where $\Gamma$ has $q_i = 3$ "sides" at $\boldsymbol{x}_i$. Functions in $V^p(\Omega_h;\Gamma)$ may admit three distinct limits at $\boldsymbol{x}_i$. The bridges $\{1,2\}$, $\{2,3\}$ and $\{1,3\}$ around $\boldsymbol{x}_i$ are signaled by gray arrows.
• Figure 3: Figure for the proof of \ref{['prop:solidangles']}. The cone $C$ is shaded in gray.
• Figure 4: A face-connected sequence of elements of $\Omega_h$ circling around a subsimplex $S$ and linking $K^-(F)$ with $K^+(F)$.
• Figure 5: A Lagrange node $\boldsymbol{x}_i$ on the hypersurface $\Gamma$ (solid black line), its star $\textup{st}(\boldsymbol{x}_i)$ (filled in gray), and the components $\textup{st}(\boldsymbol{x}_i;j)$ for $j = 1,2,3$. The gray solid lines represent the boundary of some mesh elements of $\Omega_h$.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3: Estimate with global norms
• proof
• Remark 1.4: The space $H^1(\Omega \setminus \Gamma)$
• Lemma 2.1
• proof
• Theorem 2.2: Multi-trace interpolant
• Remark 2.3
• proof : Proof of \ref{['cor:multitrace']}