Adaptative finite element simulation of currents at microelectrodes to a guaranteed accuracy. Application to a simple model problem

In this series of papers we consider the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the `edge-effectʹ. Our approach to overcoming this problem involves the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm. This allows us to calculate the current to within a prescribed tolerance. We begin by demonstrating the power of the method for a simple model problem ¯¯ an E reaction mechanism at a microdisc electrode ¯¯ for which the analytical solution is known. In this paper we give the background to the problem, and show how an a posteriori error bound can be used to drive an adaptive mesh-generation algorithm. We then use the algorithm to solve our model problem and obtain very accurate results on comparatively coarse meshes in minimal computing time. We give the technical details of the background theory and the derivation of the error bound in the accompanying paper (K. Harriman et al., Electrochem. Commun. 2 (2000) 157).