  Discrete exterior calculus (DEC) is a framework for constructing discrete
versions of exterior differential calculus objects, and is widely used in
computer graphics, computational topology, and discretizations of the
Hodge-Laplace operator and other related partial differential equations.
However, a rigorous convergence analysis of DEC has always been lacking; as far
as we are aware, the only convergence proof of DEC so far appeared is for the
scalar Poisson problem in two dimensions, and it is based on reinterpreting the
discretization as a finite element method. Moreover, even in two dimensions,
there have been some puzzling numerical experiments reported in the literature,
apparently suggesting that there is convergence without consistency. In this
paper, we develop a general independent framework for analyzing issues such as
convergence of DEC without relying on theories of other discretization methods,
and demonstrate its usefulness by establishing convergence results for DEC
beyond the Poisson problem in two dimensions. Namely, we prove that DEC
solutions to the scalar Poisson problem in arbitrary dimensions converge
pointwise to the exact solution at least linearly with respect to the mesh
size. We illustrate the findings by various numerical experiments, which show
that the convergence is in fact of second order when the solution is
sufficiently regular. The problems of explaining the second order convergence,
and of proving convergence for general p-forms remain open.
