Jump-preserving polynomial interpolation in non-manifold polyhedra

We construct a piecewise-polynomial interpolant u ↦ Π u u ↦ Π u for functions u : Ω ∖ Γ → R u:Ω Γ → R , where Ω ⊂ R d Ω ⊂ R^d is a Lipschitz polyhedron and Γ ⊂ Ω Γ ⊂ Ω is a possibly non-manifold ( d − 1 ) (d-1) -dimensional hypersurface. This interpolant enjoys approximation properties in Sobolev norms, as well as a set of additional algebraic properties, namely, Π 2 = Π Π ^2 = Π , and Π Π preserves homogeneous boundary values and jumps of its argument on Γ Γ . As an application, we obtain a bounded discrete right inverse of the “jump” operator across Γ Γ , and an error estimate for a Galerkin scheme to solve a second-order elliptic PDE in Ω Ω with a prescribed jump across Γ <mml:annotation en