  In a 1988 article, Dziuk introduced a nodal finite element method for the
Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear
triangulation, initiating a line of research into surface finite element
methods (SFEM). Demlow and Dziuk built on the original results, introducing an
adaptive method for problems on 2-surfaces, and Demlow later extended the a
priori theory to 3-surfaces and higher order elements. In a separate line of
research, the Finite Element Exterior Calculus (FEEC) framework has been
developed over the last decade by Arnold, Falk and Winther and others as a way
to exploit the observation that mixed variational problems can be posed on a
Hilbert complex, and Galerkin-type mixed methods can be obtained by solving
finite dimensional subproblems. In 2011, Holst and Stern merged these two lines
of research by developing a framework for variational crimes in abstract
Hilbert complexes, allowing for application of the FEEC framework to problems
that violate the subcomplex assumption of Arnold, Falk and Winther. When
applied to Euclidean hypersurfaces, this new framework recovers the original a
priori results and extends the theory to problems posed on surfaces of
arbitrary dimensions. In yet another seemingly distinct line of research,
Holst, Mihalik and Szypowski developed a convergence theory for a specific
class of adaptive problems in the FEEC framework. Here, we bring these ideas
together, showing convergence and optimality of an adaptive finite element
method for the mixed formulation of the Hodge Laplacian on hypersurfaces.
