A New Multilevel Method for Electrostatic Problems through Hierarchical Loop Basis

TL;DR

The paper introduces a multilevel hierarchical loop basis method for efficiently solving Poisson's equation in electrostatics, significantly reducing computation time through a novel multilevel approach.

Contribution

It extends previous loop-tree basis methods to a multilevel framework, improving convergence speed and computational efficiency in solving electrostatic problems.

Findings

Achieved faster convergence in numerical examples.

Reduced overall solution time compared to existing methods.

Demonstrated effectiveness on benchmark electrostatic problems.

Abstract

We present a new multilevel method for calculating Poisson's equation, which often arises form electrostatic problems, by using hierarchical loop bases. This method, termed hierarchical Loop basis Poisson Solver (hieLPS), extends previous Poisson solver through loop-tree basis to a multilevel mesh. In this method, Poisson's equation is solved by a two-step procedure: First, the electric flux is found by using loop-tree basis based on Helmholtz decomposition of field; Second, the potential distribution is solved rapidly with a fast solution of O(N) complexity. Among the solution procedures, finding the loop part of electric flux is the most critical part and dominates the computational effort. To expedite this part's convergent speed, we propose to use hierarchical loop bases to construct a multilevel system. As a result, the whole solution time has been noticeably reduced. Numerical examples are presented to demonstrate the efficiency of the proposed method.