AI 中文总结
该研究为对称网格上的离散外微积分算子建立块对角化框架，利用群作用证明算子等变性，实现并行求解，适用于弯曲流形，通过实验验证其在不同网格上的有效性，降低了结构保持DEC计算的线性求解成本。
AI 中文摘要
我们为对称网格上的离散外微积分（DEC）算子建立了一个通用的块对角化框架，实现了具有可证明的FLOP减少的简单并行求解器。我们证明了两个基本的DEC算子，离散外导数$d$和霍奇星算子$\star$，在单纯复形上的等距有限群作用下是等变的。该证明利用了群作用在余链空间上诱导的置换表示。因此，任何由$d$和$\star$组装而成的算子（包括霍奇拉普拉斯算子、余微分算子、麦克斯韦型算子和弹性算子）在单个对称适应基中继承了块对角结构，该基仅在每个网格上计算一次。与限于平坦柏拉图域的谱方法不同，该框架原生适用于弯曲流形，原则上适用于对称域上的计算电磁学和几何流体模拟。在测地球面（$I_h$对称性）和六边形环面（$D_{6h}$对称性）上的数值实验相对于密集直接分解分别产生了高达$62\times$和$182\times$的基于FLOP的并行加速。在具有$T_d$对称性的平坦三维环面$T^3$的体心立方（BCC）细分上的进一步实验证实了外导数、霍奇星算子和霍奇拉普拉斯算子在机器精度下对于三个网格分辨率的形式度$k = 0,1,2$的等变性。基于FLOP的顺序加速接近其理论渐近值$\approx 9.07\times$，标准的舒尔多重性减少将其进一步加深了约$|G|$倍。这些结果表明，单个对称适应基降低了弯曲和三维网格上结构保持DEC计算的线性求解成本。
英文摘要
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.
Comments33 pages, 8 figures, 6 tables. Code: https://github.com/ldsufrpe/dec-equivariance
