The solution procedure for the multi-grid finite element method is that the algebraic equations formed by the finite element method are in turn smoothed by the interpolation from the coarse grids to the fine grids and the restriction from the fine grids to the coarse grids, which is the finite

Newton-Raphson iteration. In the present investigation, a multi-grid W-cycle and the finest grids with 1025 x 1025 nodes are adopted, where the number of the layers of the W-cycle is 4.

must be assembled and solved at each

Newton-Raphson iteration. Note that m is usually either smaller or much smaller than n.

For a given small parameter [tau], we first resolve numerically the following system of w and [eta], by using the mixed Fourier-Legendre pseudospectral method suggested in the next subsection and a

Newton-Raphson iteration process with the initial data [eta] = 0 and w = 0:

As can be seen in Figure 3, more than one

Newton-Raphson iteration is needed to obtain a converged solution.

The

Newton-Raphson iteration converges within 2-3 steps given that changes are usually relatively small.

Here,

Newton-Raphson iteration method is employed, and through transposition treatment, we get the following expression:

Elements Pedersen Proposed Speed-up 1st 6 3.0144 2.4427 123.4044 2nd 216 192.5288 159.6241 120.6139 3rd 224 586.2708 612.5911 95.7034 4th 1580 2840.7558 2401.0994 118.3106 Table 11:

Newton-Raphson iteration count to reach desired accuracy (Pedersen: Pedersen's nonlinear FEM; Proposed: proposed nonlinear FEM).

Following [19], we define the profile

Newton-Raphson iteration formula as follows:

The

Newton-Raphson iteration scheme for solving algebraic equations is derived by the first-order Taylor expansion and gives a recurrence formula for the approximate solution of an equation.

Schulte, "A decimal floating-point divider using

newton-raphson iteration," Journal of VLSI Signal Processing Systems, vol.

A new registration algorithm based on

Newton-Raphson iteration is proposed to align images with rigid body transformation.

The system of nonlinear melt flow equations is solved via a

Newton-Raphson iteration that facilitates efficient computation of the design sensitivities.