A novel iterative method for discrete Helmholtz decomposition

摘要

A new iterative method for the computation of the discrete Helmholtz decomposition of a vector is presented. We are particularly interested in computing the discrete Helmholtz decomposition when the given vector is discretized by a mixed finite element method defined by Raviart-Thomas (RT) or Brezzi-Douglas-Marini (BDM) elements. The decomposition is computed by solving a system of linear equations by an iterative method, that splits a given vector into a divergence-free component and a curl-free component. Each iteration cycle uses a well-developed solver based on the algebraic multigrid method for computing a projection onto H(div) or H(curl). Only a few iteration cycles are required to compute an accurate approximate solution. As a by-product, we obtain an iterative method for the solution of linear systems of equations with a nearly singular matrix.