The h-p method is shown to be a powerful tool for exhibiting the presence of a singularity at the extremity of a liquid-liquid interface without surface tension, in a Newtonian isothermal layered flow. Numerical results indicate that no pressure continuity seems to hold on the interface separation line.
Levieux, F., Berghezan, D., & Dupret, F. (1994). Analysis of the Interface Singularity of a 2-fluid Flow By H and H-p Finite-elements. Computer Methods in Applied Mechanics and Engineering, 116(1-4), 165-174. https://doi.org/10.1016/S0045-7825(94)80020-0 (Original work published 1994)