Guzmán et al. explain their method with a polygonal, convex, tubular domain, and an immersed interface definable with a function with the first derivative, which does not change in time. The jumps across a discontinuity are defined as expressions dependent on density and velocity, before and after the interface. Local basis functions are defined for the interface triangles that are cut by the interface. An additional stabilization term is added when the density contrast ρ^-/ρ⁺ is high. Next, the authors prove that error estimates are indeed independent of such contrast. The basis functions for the interface are penalized only in the normal derivative jumps, but not in the tangential derivative jumps.

The authors show that their method has optimal local approximation properties, using, among other lemmas, Poincaré’s inequalities. They also prove a priori and L² error estimates. Finally, an extension to three dimensions is mentioned, assuming the interface is a simple C² surface; also, continuity may be enforced through, alternatively, defining two points of the interface that intersect part of the boundary, extendable to Cartesian grids. Examples of implementation results are shown.