A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions

Abstract

We propose a novel efficient algorithm to solve Poisson equation in irregular two dimensional domains for electrostatics. It can handle Dirichlet, Neumann or mixed boundary problems in which the filling media can be homogeneous or inhomogeneous. The basic idea of the new method is solve the problem in three steps: (i) First solve the equation ∇ · D = ρ. The inverse of the divergence operator in a restricted subspace is found to yield the electric flux density D by a fast direct solver in O(N ) operations. The D so obtained is nonunique with indeterminate divergence-free component. Then the electric field is found by E = D/ǫ. But ∇ × E = 0 for electrostatic field; hence, E is curl free and orthogonal to the divergence free space. (ii) An orthogonalization process is used to purify the electric field making it curl free and unique. (iii) Then the potential φ is obtained by solving ∇φ = −E or finding the inverse of the gradient operator in a restricted subspace by a similar fast direct solver in O(N ) operations. Treatments for both Dirichlet and Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, it is observed that the computational complexity of our proposed method almost scales as O(N ), where N is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

A new multilevel method for electrostatic problems through hierarchical loop basis

“…The Poisson solver through looptree basis has demonstrated its efficiency in [18]. Here, we further compare the difference between the proposed method and traditional nodal basis FEM in terms of basis function space.…”

“…A thorough description of handling these two problems can be found in [18]. The repetitious details need not be given here.…”

“…Recently, a novel Poisson solver has been proposed to solve 2D problems [17,18]. In contrast to traditional Poisson solvers, this method solves the electric flux D directly.…”

Self-gravity in curved mesh elements

“…While the Poisson equation appears as the privileged alternative, it requires the knowledge of accurate boundary conditions that only the integral approach can provide, except in the very specific case of periodic boundary conditions. The gradual increase in the precision of models and simulations is a considerable source of motivation to derive new formulae, and explore various techniques and algorithms to solve this problem more and more efficiently (Grandclément et al, 2001;Matsumoto and Hanawa, 2003;Huré, 2005;Jusélius and Sundholm, 2007;Li et al, 2008;Guillet and Teyssier, 2011;Ma et al, 2012).…”