We develop a finite element method for elliptic partial differential
equations on so called composite surfaces that are built up out of a finite
number of surfaces with boundaries that fit together nicely in the sense that
the intersection between any two surfaces in the composite surface is either
empty, a point, or a curve segment, called an interface curve. Note that
several surfaces can intersect along the same interface curve. On the composite
surface we consider a broken finite element space which consists of a
continuous finite element space at each subsurface without continuity
requirements across the interface curves. We derive a Nitsche type formulation
in this general setting and by assuming only that a certain inverse inequality
and an approximation property hold we can derive stability and error estimates
in the case when the geometry is exactly represented. We discuss several
different realizations, including so called cut meshes, of the method. Finally,
we present numerical examples
