Isoparametric formulation for a quadrilateral element described by non-differentiable functions using Gaussian points

The isoparametric formulation for mapping from natural (intrinsic) coordinates to global coordinates is performed to facilitate point placement for solving differential equations using numerical methods. Here, we attempt to derive the general form of this formulation for a quadrilateral so that these shapes can be considered as an image of the reference square Ωn=[−1,1]×[−1,1] in natural coordinates with desired accuracy. Since this formulation is based on Lagrange interpolation, its error analysis is conducted based on the type of fixed points chosen. It is afterwards shown that, contrary to common belief, Legendre–Gauss points, which do not include the endpoints of the natural interval, can also be used for interpolation-based mapping. To the best of our knowledge, the application of Legendre–Gauss points in this specific context has not been previously reported. Additionally, using a practical example and its precise mapping, we demonstrate that in view of the limitations of triangular mapping, any arbitrary shape in global coordinates—even those with sides described by non-differentiable or piecewise functions—can be considered as a projection of the reference square. This, along with the definition domain of two-dimensional spectral methods as the square [−1,1]×[−1,1], highlights the advantage and superiority of the isoparametric formulation for the quadrilateral element.