# Anisotropy in 2D Discrete Exterior Calculus

## TL;DR

This paper introduces a local formulation for 2D Discrete Exterior Calculus that effectively handles material heterogeneity and anisotropic fluxes, demonstrating comparable or improved accuracy over FEML in solving anisotropic Poisson equations.

## Contribution

It presents a novel local formulation of 2D DEC that incorporates anisotropy and heterogeneity, aligning its computational cost with FEML while improving solution accuracy.

## Key findings

- DEC solutions converge numerically for anisotropic Poisson problems.
- DEC solutions are comparable to FEML on fine meshes.
- DEC performs slightly better than FEML on coarse meshes.

## Abstract

We present a local formulation for 2D Discrete Exterior Calculus (DEC) similar to that of the Finite Element Method (FEM), which allows a natural treatment of material heterogeneity (element by element). It also allows us to deduce, in a robust manner, anisotropic fluxes and the DEC discretization of the pullback of 1-forms by the anisotropy tensor, i.e. we deduce how the anisotropy tensor acts on primal 1-forms. Due to the local formulation, the computational cost of DEC is similar to that of the Finite Element Method with Linear interpolations functions (FEML). The numerical DEC solutions to the anisotropic Poisson equation show numerical convergence, are very close to those of FEML on fine meshes and are slightly better than those of FEML on coarse meshes.