In recent years, researchers focused on the analysis of the Lagrangian Coherent Structure (LCS) in complex flows owing to its accurate description of flow field details. In this work, a novel and accurate numerical technique based on Lagrangian-Eulerian stabilized collocation method (LESCM) is proposed to compute Finite Time Lyapunov Exponents (FTLEs) used for extracting LCSs in viscous incompressible flows. LESCM, which has been used for simulating the fluid-structure interaction problem [1] was developed based on the material point method (MPM). Since the hybrid Lagrangian-Eulerian description is employed in the LESCM, the trajectory of fluid flow can be explicitly tracked throughout the simulation, which enables the direct evaluation of the FTLE field. And thanks to the Eulerian characteristic of LESCM, errors of FTLEs caused by the particle shifting technique (PST) can be completely avoided since the deformation gradient used for computing FTLEs is calculated based on the fixed Eulerian nodes rather than the fluid particles with unphysical shifting. Thus, the novel technique based on LESCM outmatches the pure Lagrangian particle methods in accuracy and provides an accurate way to detect complex LCSs in flow fields. Furthermore, due to the high efficiency of the LESCM, up to 10 million particles can be used easily to detect LCSs without the need for parallel computation techniques. Several numerical examples including flow passing a circular cylinder and swimming fishes in flowing water are tested, which demonstrates the effectiveness of the novel approach under various conditions.