  In this paper we will discuss different coupling methods {suitable for use
in} the framework of the recently introduced CutFEM paradigm, cf. Burman, Erik;
Claus, Susanne; Hansbo, Peter; Larson, Mats G.; Massing, Andr\'e . CutFEM:
discretizing geometry and partial differential equations. Internat. J. Numer.
Methods Engrg. 104 (2015), no. 7, 472-501. In particular we will consider
mortaring using Lagrange multipliers on the one hand and Nitsche's method on
the other. For simplicity we will first discuss these method in the setting of
uncut meshes, and end with some comments on the extension to CutFEM. We will,
for comparison, discuss some different types of problems such as high contrast
problems and problems with stiff coupling or adhesive contact. We will review
some of the existing methods for these problems and propose some alternative
methods resulting from crossovers from the Lagrange multiplier framework to
Nitsche's method and vice versa.
