For the solution of A-EFIE matrix equation, an

iterative solver is often used because the underlying matrix-vector product (MVP) can be accelerated by fast algorithms such as the fast multipole algorithm (FMA) [15,18-21].

In particular, for a two-grid approach, i.e., l = 1,2, one can describe the realization as follows: the method performs a certain number [v.sub.1] of smoothing steps using an

iterative solver that can be, for instance, a weighted Jacobi, a Gauss-Seidel, or a Krylov subspace method like GMRES [30, 31]; the residual of the current iterate is computed and restricted by a matrix-vector multiplication with the restriction matrix Q [member of] [mathematical expression not reproducible]; the operator [A.sub.1] = Ais restricted via a Petrov-Galerkin construction to obtain the coarse-grid operator, [A.sub.1] = Q[A.sub.1]P [MEMBER OF] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE], where P [member of] [R.sup.ncxn] is the interpolation operator.

Meanwhile, the 2D models are solved by both a direct and an

iterative solver to study the effect on the computational efficiency of the decoupling approach compared to different solvers.

However, iterative methods still have some significant disadvantages compared with direct methods: (1) The iteration number for an

iterative solver is highly sensitive to the conditioning of the system matrix.

Multi-grid method works by decomposing a problem into separate length scales, and using an

iterative solver method that optimizes error reduction for that length scale.

For each simulation 3D periodic boundary conditions and the Marc

iterative solver are used.

Malony, "A 3D vector-additive

iterative solver for the anisotropic inhomogeneous poisson equation in the forward EEG problem," in Computational Science--ICCS 2009, vol.

Soleymani, "A fast convergent

iterative solver for approximate inverse of matrices," Numerical Linear Algebra with Applications, vol.

Because of numerical instability, the

iterative solver with fixed relaxation requires the most simulation time, which is many times greater than other methods, especially the semi-simultaneous solver.

The finite element method simulator features CPU multithreading enhancements to the

iterative solver.

Momentum automatically chooses the appropriate solver for a given problem, depending on the problem size (number of unknowns), some other properties of the EM problem (especially ease and effectiveness of precondition for the

iterative solver) and the platform/machine on which Momentum is running.