Parallelisation of Discrete Exterior Calculus via Representation Theory on Curved and Three-Dimensional Meshes

Abstract

We establish a universal block-diagonalization framework for Discrete Exterior Calculus (DEC) operators on symmetric meshes, enabling embarrassingly parallel solvers with provable FLOP reductions. We prove that the two fundamental DEC operators, the discrete exterior derivative $d$ and the Hodge star $\star$, are equivariant under isometric finite group actions on simplicial complexes. The proof exploits the permutation representation induced on cochain spaces by the group action. As a consequence, any operator assembled from $d$ and $\star$ (including the Hodge Laplacian, the codifferential, Maxwell-type operators, and elasticity operators) inherits a block-diagonal structure in a single symmetry-adapted basis, which is computed only once per mesh. Unlike spectral methods restricted to flat Platonic domains, the framework applies natively to curved manifolds and is applicable in principle to computational electromagnetism and geometric fluid simulation on symmetric domains. Numerical experiments on a geodesic sphere ($I_h$ symmetry) and a hexagonal torus ($D_{6h}$ symmetry) yield FLOP-based parallel speedups, relative to a dense direct factorization, of up to $62\times$ and $182\times$, respectively. A further experiment on a body-centred-cubic (BCC) tessellation of the flat 3-torus $T^3$ with $T_d$ symmetry confirms equivariance of the exterior derivative, Hodge star, and Hodge Laplacian at machine precision for form degrees $k=0,1,2$ across three mesh resolutions. The FLOP-based sequential speedup approaches its theoretical asymptote of $\approx 9.07\times$, which a standard Schur-multiplicity reduction deepens by a further factor of order $|G|$. These results show that a single symmetry-adapted basis reduces the linear-solve cost of structure-preserving DEC computations on curved and three-dimensional meshes.