Terms involving jumps of stresses on boundaries are proposed for the finite element approximation of the Stokes problem and the linear elasticity equations. These terms are designed to improve the transmission conditions between subdomains at three different levels, namely, between the element domains, between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid–structure interaction problems. The benefits in each case are respectively the possibility of using discontinuous pressure interpolations in a stabilized finite element approximation of the Stokes problem, a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behavior of iterative schemes to couple the fluid and the solid in fluid–structure integration algorithms. The motivation to introduce these terms stems from a decomposition of the unknown into a conforming and a non‐conforming part, a hybrid formulation for the latter and a simple approximation for the unknowns involved in the hybrid problem.