Projection stabilisation of Lagrange multipliers for the imposition of constraints on interfaces and boundaries

Projection stabilisation applied to general Lagrange multiplier finite element methods is introduced and analysed in an abstract framework. We then consider some applications of the stabilised methods: (i) the weak imposition of boundary conditions, (ii) multiphysics coupling on unfitted meshes, (iii) a new interpretation of the classical residual stabilised Lagrange multiplier method introduced in H. J. C. Barbosa and T. J. R. Hughes, The finite element method with Lagrange multipliers on the boundary: circumventing the Babu\v{s}ka-Brezzi condition, Comput. Methods Appl. Mech. Engrg., 85(1):109--128, 1991 .