Mimetic framework on curvilinear quadrilaterals of arbitrary order

TL;DR

This paper introduces higher order mimetic discretizations on curvilinear quadrilaterals that preserve geometric and topological properties, ensuring consistency between continuous and discrete differential forms.

Contribution

It develops a novel framework linking differential geometry and algebraic topology for mimetic spectral elements with orientation preservation and Hodge decomposition.

Findings

Discrete operators preserve orientation.

The framework ensures commutation between continuous and discrete operations.

Hodge decomposition is extended to discrete and finite-dimensional settings.

Abstract

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.