The mortar element method on composite meshes

The mortar element method (MEM) is a non-conforming domain decomposition technique, introduced by Bernardi, Maday and Patera [2] in the late 80s. It allows to couple different variational discretization approaches for PDEs, where different has to be intended in a large sense. Namely, it is possible either to have dfferent independent meshes in the subdomains of the decomposition or to couple finite elements in some subdomains with spectral ones in others or to couple different approximation degrees in the subdomains (and all possible combinations of these configurations).
Originally, the MEM was intended for non-overlapping subdomains. Later, to enlarge its applicability and flexibility, we studied a generalization of this method on composite meshes (thus with overlapping subdomains, but different from that proposed by Achdou and Maday [1]). This new mortar element method on overlapping meshes have been welcomed in several contexts, as eddy current non-destructive testing and free-boundary axisymmetric plasma equilibria in realistic geometries.Two meshes can either fully overlap or partially overlap in a narrow region. This approach allows to deal with the "random" movement of one domain w.r.t. the other or to achieve easily higher order regularity for the approximated fields while preserving accurate meshing of the geometric details. The continuity of the numerical solution in the region of overlap is weakly enforced by a suitable L² projection.
We refer to for the interested reader.

References
[1] Y. Achdou, Y. Maday - The mortar element method with overlapping
subdomains. SIAM J. Numer. Anal., vol. 40, no. 2 (2002) 601-628.
[2] C. Bernardi, Y. Maday, A. Patera - A new non-Conforming approach
to domain decomposition: the mortar element method. Seminaire XI du College
de France, edite par Brezis, H., Lions, J. dans Nonlinear partial di erential
equations and their applications, Pitman, (1994) 13-51.
[3] See "Non-conforming FEM or SEM" section at the web page
https://math.unice.fr/ ~frapetti/pub1.html