Jump-preserving polynomial interpolation in non-manifold polyhedra

2022/11/15 by Averseng, Martin
#41A10 #65N12 #65N15 #65N38 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2211.08223

Abstract

We construct a piecewise-polynomial interpolant u ↦ Πu for functions u:Ω∖ Γ→ ℝ, where Ω⊂ ℝ^d is a Lipschitz polyhedron and Γ⊂ 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, Π² = 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.

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