A new multilevel method for electrostatic problems through hierarchical loop basis

Abstract

a b s t r a c tWe present a new multilevel method for calculating Poisson's equation, which often arises from electrostatic problems, by using hierarchical loop basis. This method, termed as 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 time. To expedite this part's convergent speed, we propose to use hierarchical loop basis 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.

Point-Symmetric Extension-Based Interval Shannon-Cosine Spectral Method for Fractional PDEs

“…Sinc is a famous sampling function [20] based on approximating the Dirac delta function as a band-limited function and is given by…”

“…In recent years, wavelet analysis theory has been developed to be a powerful tool to solve the fractional partial differential equations in recent years [19][20][21]. Shannon wavelets have been constructed based on the sinc function.…”