A mathematical dipole is widely used as a model for the primary current source in electroencephalography (EEG) source analysis. In the governing Poisson-type differential equation, the dipole leads to a singularity on the right-hand side, which has to be treated specifically. In this paper, we will present a full subtraction approach where the total potential is divided into a singularity and a correction potential. The singularity potential is due to a dipole in an infinite region of homogeneous conductivity. The correction potential is computed using the finite element (FE) method. Special care is taken in order to evaluate the right-hand side integral appropriately with the objective of achieving highest possible convergence order for linear basis functions. Our new approach allows the construction of transfer matrices for fast computation of the inverse problem for anisotropic volume conductors. A constrained Delaunay tetrahedralisation (CDT) approach is used for the generation of high-quality FE meshes. We validate the new approach in a four-layer sphere model with a highly conductive cerebrospinal fluid (CSF) and an anisotropic skull compartment. For radial and tangential sources with eccentricities up to 1 mm below the CSF compartment, we achieve a maximal relative error of 0.71% in a CDT-FE model with 360 k nodes which is not locally refined around the source singularity and therefore useful for arbitrary dipole locations. The combination of the full subtraction approach with the high quality CDT meshes leads to accuracies that, to the best of the author's knowledge, have not yet been presented before.