In the present study, the enriched finite element method is applied to compute the intensity of singularity for three-material joints with power-logarithmic singularities. The method is modified by using a higher order polynomial function, i.e., changing an element type from linear to quadratic element. Different sizes of the enriched region and mesh refinement technique are used to study its convergence. Furthermore, the models with various lengths are investigated to study an influence of geometry on the singular stress field.