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Abstract: It has long been established that when a small-displacement problem involving contact between two or more bodies is solved, conforming algorithm approaches based on BEM are reliable and robust. Nevertheless, the authors have shown in a previous paper that a relative sliding between the bodies in contact, even if small, can originate a fictitious corner in the numerical model, which may alter the stress and the displacement predictions. This problem disappears if a non-conforming algorithm is employed. In this paper a new problem with different features but also presenting artificial alteration of the results is addressed. The compression of a rounded punch on a foundation is analysed for many fillet radii. Conforming and non-conforming algorithms are used. Comparison of results shows that effects of an artificial ‘numerical corner’ may alter even the traction distribution relatively far from it. Consequently, the assumed connection between small displacements and initially conforming discretizations with the use of a conforming approach disappears. An adequate knowledge of the contact problem under consideration is then essential to determine whether the singularity is inherent to the problem or is artificial, a non-conforming algorithm then being recommended in this second case. [Copyright &y& Elsevier]