In recent years, several alternative non-standard finite element methods (FEMs) have been developed to improve the accuracy and convergence of the standard FEM. One of the proposed alternative FEMs which is of our interest is a method that combines the FEM and meshfree method using the partition of unity concept, called the three-node triangular element with continuous nodal stress (T3-CNS). In the T3-CNS element formulation, the shape functions are constructed using a combination of finite element continuous nodal gradient shape functions and a set of mesh-free shape functions obtained using the orthonormalized and constrained least-squares method. The aim of this paper is to present a numerical study on the accuracy and convergence of the T3-CNS interpolation when approximating several mathematically defined surfaces and their gradients. In addition, the discontinuous nodal stress version of the element, called the T3-DNS, is examined. The results are compared to those obtained using the standard triangular element and the Kriging-based triangular element. The results show that both the T3-CNS and T3-DNS interpolations possess the consistency property, providing highly accurate surface fittings, and exhibiting excellent convergence. Therefore, both the T3-CNS and T3-DNS interpolations are suitable to be employed as the trial function in numerical methods based on the Rayleigh-Ritz or Galerkin method.