In this work, triangular C1 interpolation based on Argyris element (AE) is used to develop the ARgyris element based searching (ARES) algorithm, an efficient tool for the accurate identification of stationary points (SPs), e.g. X-point, magnetic axis, snowflake point, etc and separatrix in magnetic confinement fusion application. AE-based interpolation allows the formulation of the nonlinear problem of SP identification on the triangle as a vectorial root-finding problem, addressed via different Newton-like algorithms. The proposed method is able to detect both first-order SP (e.g. X-points and magnetic axis, where the norm of the gradient vanishes) and second-order SP (e.g. snowflake points, where both the norm of the gradient and of the second derivatives vanishes). The separatrix is detected inside the triangles via the novel zero line searching algorithm, and is described by means of quintic spline interpolant of its components. Thanks to its modularity and efficiency, ARES is suitable to run on structured and unstructured triangular meshes, and can be readily integrated with any kind of Grad–Shafranov solver based on 2D triangular finite element method, or used for real-time SP and separatrix identification for control oriented purposes.