Abstract: Due to the surface roughness effect in real engineering, contact loads often have a certain randomness and complexity. Numerical approach is the chosen method for effectively and accurately handling the pertaining contact problems. The piecewise constant discretization scheme has been widely used because of its simplicity. In the two-dimensional contact analyses, it will inevitably incur infinite discontinuities in the displacement gradients between adjacent elements, possibly leading to issues of numerical instability or robustness . In the present paper, a refined scheme featuring uniform discretization using overlapping triangles is proposed. Compared with the primitive schemes based on either concentrated force or piecewise constant pressure discretization, overlapping triangular element can fit the distributed contact pressure more effectively, and hence achieves a superb numerical accuracy and convergence. It is noted that the overlapping triangular discretization has been attempted in JOHNSON’s classical monographs, which however managed to handle just dozens of elements. We demonstrate that the triangular discretization can be seamlessly incorporated with the Fast Fourier Transforms (FFT) techniques to achieve a tremendous refinement, which makes it possible to deal with millions of elements using a personal computer. In the present work, the elementary solutions to the triangular loading applied on an elastic layer, either perfectly bonded to or frictionlessly rested on a rigid substrate, are presented for ease of FFT programming. The construction of a cyclic matrix for implementing the DC-FFT algorithm is discussed. Benchmark examples are examined and compared against the finite element results.