Dual Discrete Geometric Methods in Terms of Scalar Potential on Unstructured Mesh in Electrostatics

Dual formulations established on dual unstructured meshes using the discrete geometric method (DGM) for electrostatic field problems are presented. The formulations are both in terms of scalar potential. When compared with traditional dual formulation in terms of vector potential, the proposed method is more efficient with reduced number of unknowns and alleviated computational complexity. The positive definiteness of the constitutive matrices requires the dual meshes satisfy the Voronoi-Delaunay condition. For the elements not satisfying this condition, an approximate element level diagonalization method is proposed. The complementary bounds of electrical energy are investigated through a micro-electro-mechanicals system comb driver example. A comparison between the DGM and the finite-element method is performed.