In this paper higher order mimetic discretizations are introduced which are
firmly rooted in the geometry in which the variables are defined. The paper
shows how basic constructs in differential geometry have a discrete counterpart
in algebraic topology. Generic maps which switch between the continuous
differential forms and discrete cochains will be discussed and finally a
realization of these ideas in terms of mimetic spectral elements is presented,
based on projections for which operations at the finite dimensional level
commute with operations at the continuous level. The two types of orientation
(inner- and outer-orientation) will be introduced at the continuous level, the
discrete level and the preservation of orientation will be demonstrated for the
new mimetic operators. The one-to-one correspondence between the continuous
formulation and the discrete algebraic topological setting, provides a
characterization of the oriented discrete boundary of the domain. The Hodge
decomposition at the continuous, discrete and finite dimensional level will be
presented. It appears to be a main ingredient of the structure in this
framework.Comment: 69 page