Interdisciplinary Applied Mathematics

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—аф (ф*)n cosh(a (ф*)п),

where (n) denotes the iteration number. The solution from the previous iteration is used for evaluation of the nonlinear forcing function, and the resulting system is solved until the residual is reduced beyond a certain level (typically 10~13).

The numerical solution of equation (7.4) is challenging due to the exponential nonlinearity associated with the hyperbolic sine function. In particular, for large values of a the nonlinear forcing increases rapidly for any value of ф. Also, for very large values of ф with a = 1, similar difficulties exist. Accurate resolution of the problem requires high grid density within the EDL. A typical mesh for the a =1 and ф = 10,000 case is presented in Figure 14.4 (right). It consists of 22 elements across the channel width, spaced in biased fashion with minimum width of 0.001h very near the walls. Once the mesh topology is fixed, the modal expansion order N is increased to resolve the problem further. For rectangular elements shown in Figure 14.4, N = 2 corresponds to a quadratic solution for ф*, typically employed in finite element discretizations.

FIGURE 14.4. EDL resolution: Exponential decay of the L2 error norm as a function of the spectral expansion order N (left). Sample grid used to resolve sharp electric double layer consists of 22 elements across the channel, and each element is discretized with Nth-order modal expansion per direction (right). The quadrature points for su-elemental discretization at select elements are also shown. Simulations are performed for в = 10,000.

The numerical accuracy of the results is determined by using equation (7.9). In Figure 14.4 (left) the variation of the L2 error norm as a function of the modal expansion order N is presented. The results are obtained for the mesh topology shown on the right plot. Convergence results for a = 1, в = 10,000 and a = 10, в = 10,000 are shown. The L2 error norm is defined as

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