A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions (Communications in Computational Physics, 2020)
Even though there are various fast methods and preconditioning techniquesavailable for the simulation of Poisson problems, little work has been done for solvingPoisson's equation by using the Helmholtz decomposition scheme. To bridge thisissue, we propose a novel efficient algorithm to solve Poisson's equation in irregulartwo dimensional domains for electrostatics through a quasi-Helmholtz decompositiontechnique – the loop-tree basis decomposition. It can handle Dirichlet, Neumann ormixed boundary problems in which the filling media can be homogeneous or inhomogeneous.A novel point of this method is to first find the electric flux efficiently byapplying the loop-tree basis functions. Subsequently, the potential is obtained by findingthe inverse of the gradient operator. Furthermore, treatments for both Dirichletand Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, itis observed that the computational complexity of our proposed method almost scalesas $\mathcal{O}$$(N)$, where $N$ is the triangle patch number of meshes. Consequently, this newalgorithm is a feasible fast Poisson solver.