We construct a piecewise-polynomial interpolant u ↦ Π u u u for functions u: Ω ∖ Γ → R u: R, where Ω ⊂ R d Rᵈ is a Lipschitz polyhedron and 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 ² =, 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 <mml: annotation en
Martin Averseng (Thu,) studied this question.