Interpolation estimate for a finite-element space with embedded discontinuities

We consider a recently proposed finite-element space that consists of piecewise affine functions with discontinuities across a smooth given interface Γ (a curve in two dimensions, a surface in three dimensions). Contrary to existing extended finite element methodologies, the space is a variant of the standard conforming ${\mathbb P_1}$ space that can be implemented element by element. Further, it neither introduces new unknowns nor deteriorates the sparsity structure. It is proved that, for u arbitrary in ${W^{1,p}(\Omega \setminus \Gamma )\, \cap \, W^{2,s} (\Omega \setminus \Gamma),}$, the interpolant ${\mathcal I_hu}$ defined by this new space satisfies $${\parallel u - \mathcal I_hu\parallel L^q(\Omega ) \leqslant C \left [h^{1+\frac{1}{q}-\frac{1}{p}} \mid u\mid W^{1,p}(\Omega \setminus \Gamma) + h^2\mid u\mid W^{2,s}(\Omega \setminus \Gamma ) \right ],}$$ where h is the mesh size, ${\Omega \subset \mathbb R^d}$ is the domain, ${p > d, p \geqslant q, s \geqslant q}$ and standard notation has been adopted for the function spaces. This result proves the good approximation properties of the finite-element space as compared to any space consisting of functions that are continuous across Γ, which would yield an error in the ${L^q (\Omega)}$-norm of order ${h^{\frac{1}{q} - \frac{1}{p}}}$. These properties make this space especially attractive for approximating the pressure in problems with surface tension or other immersed interfaces that lead to discontinuities in the pressure field. Furthermore, the result still holds for interfaces that end within the domain, as happens for example in cracked domains.