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We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML). Contents 1. Introduction 1 2. 2D Exterior Differential Calculus as Vector Calculus 2 2.1. Wedge product for vectors in R 2 2 2.2. Hodge star operator for vectors in R 2 4 2.3. The Laplacian 4 2.4. Duality in Green's theorem 5 3. Discrete Exterior Calculus 6 3.1. Boundary operator 6 3.2. Dual mesh 9 3.3. Boundary operator on the dual mesh 10 3.4. Discrete Hodge star 11 3.5. DEC applied to 2D Poisson's equation 13 4. DEC for general triangulations 13 4.1. Dual mesh of an arbitrary triangle 13 4.2. Dual mesh of a general triangulation 14 5. Numerical examples 15 5.1. First example 15 5.2. Second example 17 6. Conclusions 19 References 19

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.

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