  In this paper we develop the isogeometric B\'ezier dual mortar method. It is
based on B\'ezier extraction and projection and is applicable to any spline
space which can be represented in B\'ezier form (i.e., NURBS, T-splines,
LR-splines, etc.). The approach weakly enforces the continuity of the solution
at patch interfaces and the error can be adaptively controlled by leveraging
the refineability of the underlying dual spline basis without introducing any
additional degrees of freedom. We also develop weakly continuous geometry as a
particular application of isogeometric B\'ezier dual mortaring. Weakly
continuous geometry is a geometry description where the weak continuity
constraints are built into properly modified B\'ezier extraction operators. As
a result, multi-patch models can be processed in a solver directly without
having to employ a mortaring solution strategy. We demonstrate the utility of
the approach on several challenging benchmark problems. Keywords: Mortar
methods, Isogeometric analysis, B\'ezier extraction, B\'ezier projection
