Discretisation in the Mapped Space

the five mapping functions for the disk electrode are presented, as well as the form that the diffusion equation for the disk electrode takes in the mapped spaces. We assume that the cylindrical coordinates, time and concentrations have all been normalised by the disk radius as in (12.14). The way to achieve these transformations is described in Appendix B. 

Michael et al. [148] used the mapping function used earlier by Saito [31], transforming to elliptic coordinates [269], 

This will be called MWA here. The transformation formula is in fact very old, and can be found in such sources as the texts by Sneddon [270], Morse and Fes-hbach [269] and Tranter [36]. The transformation is also used for band electrodes, with R replaced by X, measured as a distance from the centre of the band, across the band [13]. It results, in the case of the disk or band electrode, in a new diffusion equation, whose form is deferred to a later place, below. 

The MWA map has what might be regarded as a drawback. We wish to contain the concentration field that varies during the time Tmax of the simulation, that is, to have a computational domain extending over distances of about 6 sqrt(r ,a5c) from the electrode surface. This translates, upon conformal mapping, to a certain maximum value, Pmca- How this is calculated is described in Appendix D. The point is that such a calculation must be made, which however also applies to the other transformations, as will be seen below. 

The next conformal map to be developed was that of Amatore and Fosset [216], here to be called AF  

We wish to simulate by discretising on an equally spaced grid in the transformed space, and this grid should place points optimally in the original (R, Z) space. That is, they should be closely spaced near the disk edge, 

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