We describe how a discontinuous Galerkin finite element method with interior penalty can be used to compute the current flowing at a microdisc electrode for the steady-state E and EC' reaction mechanisms and at a channel microband electrode for the steady-state E, ECE, EC2E, DISP1 and DISP2 reaction mechanisms. We then state an a posteriori error bound for the computed current and use this as the basis of an hp-adaptive discontinuous Galerkin finite element algorithm to deliver the current to within a prescribed error tolerance. For all problems we demonstrate excellent agreement with previous results using comparatively coarse meshes. (c) 2005 Elsevier B.V. All rights reserved.