We have found that starting the iteration on a coarse mesh has several important advantages. One is that the Newton iteration is more likely to converge on a coarse mesh than on a fine mesh. Moreover, the number of variables is small on a coarse mesh and thus the cost per iteration is relatively small. Since the iteration begins from a user-specified "guess" at the solution, it is likely that many iterations will be required. Ultimately, of course, to be accurate, the solution must be obtained on a fine mesh. However, as the solution is computed on each successively finer mesh, the starting estimates are better, since they come from the converged solution on the previous coarse mesh. In general, the solution on one mesh lies within the domain of convergence of Newton’s method on the next finer mesh. [122] Thus, even though the cost per iteration is increasing, the number of required iterations is decreasing.

The adaptive placement of the mesh points to form the finer meshes is done in such a way that the total number of mesh points needed to represent the solution accurately is minimized. Specifically, we place the mesh points in the following way. To resolve the gradients we bound the variation in the solution components between mesh points as

(16–19)

and to resolve the curvature in the solution we bound the variation in the solution’s derivatives between mesh points by

(16–20)

We evaluate the above expressions between each of the mesh points. In each of the subintervals where the inequality is not satisfied, a new mesh point is placed at the midpoint of the subinterval. The parameters and are user-specified input parameters (Gradient and Curvature ratios, respectively). We exclude variables that are below a certain floor value from consideration in the adaptation. This avoids adapting on variables that are essentially zero, but due to rounding errors may show locally high derivatives. The starting estimate for the dependent variable vector is determined by a linear interpolation of the coarse mesh solution onto the new finer mesh. After determining a converged solution on this new fine mesh, the adaptation procedure is performed once again. A sequence of solutions on successively finer meshes is computed until the inequalities in Equation 16–19 and Equation 16–20 are satisfied between all mesh points.