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 simul ation of the currents at microelectrodes using an adaptive finite element a pproach. Microelectrodes typically consist of an electrode embedded (or rec essed) in an insulating material. For all such electrodes, numerical simula tion is made difficult by the presence of a boundary singularity at the ele ctrode edge ( where the electrode meets the insulator), manifested by the l arge increase in the current density at this point, often referred to as th e 'edge-effect'. Our approach to overcoming this problem involves the deriv ation 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 algor ithm. This allows us to calculate the current to within a prescribed tolera nce. We begin by demonstrating the power of the method for a simple model p roblem - an E reaction mechanism at a microdisc electrode - for which the a nalytical solution is known. In this paper we give the background to the pr oblem, and show how an a posteriori error bound can be used to drive an ada ptive mesh-generation algorithm. We then use the algorithm to solve our mod el 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). (C) 2000 Elsevier Sci ence S.A. All rights reserved.