  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.
