Micro-scale bioimpedance systems are widely used to measure the electrical properties of biological samples. While finite element simulations can accurately capture the global electric potential, they often fail to resolve boundary potential gradients, leading to inaccurate impedance estimates. This stems from sharp electric field variations at interfaces between Dirichlet electrode boundaries and Neumann insulating boundaries. Combined with microscale dimensions and complex geometries, this requires highly refined meshes near electrode edges, substantially increasing computational cost. We develop a numerically efficient method to evaluate impedance in these configurations. Near the Dirichlet–Neumann junction point on a straight boundary, the solution admits an asymptotic expansion as the distance to the junction tends to zero. In local coordinates, the differential operator is decomposed into the Laplacian plus a series of lower-order differential terms. The leading behavior of the expansion is governed by the Laplace operator and described by boundary singular functions associated with the Dirichlet–Neumann transition. Lower-order terms generate additional contributions, referred to as shadow singular functions. The coefficients of the asymptotic expansion are computed using the dual singular functions method, which allows the singular contributions to be extracted accurately without local mesh refinement. This approach captures the singular behavior of the gradient on the electrode and enables an accurate evaluation of the current, and therefore the impedance. We study and validate the method both theoretically and numerically on a simplified configuration with two planar micro-electrodes. For this geometry, standard finite element simulations require hundreds of seconds to achieve an accurate current evaluation, whereas our method achieves comparable precision in under one second. The method can be extended to multi-electrode systems, where it is expected to deliver higher accuracy while keeping computational costs low. These extensions will be explored in future work.