High-order visualization of three-dimensional Lagrangian coherent structures with DG-FTLE. The two-dimensional DG-FTLE algorithm for computing finite-time Lyapunov Exponent (FTLE) fields developed in [the authors, J. Comput. Phys. 295, 65--86 (2015; Zbl 1349.76244)] is extended to and tested in three-dimensions and for complex geometries with curved boundaries. In three-dimensions, flow deformation maps, whose eigenvalues determine the FTLE, develop steep gradients over shorter finite times than in two-dimensions. Higher-order DG approximations of the steep gradients yield Gibbs oscillations and poor conditioning of interpolation matrices. In order to reduce Gibbs oscillations, an exponential filter is used that smoothens FTLE fields. $h$-adaptivity is shown to improve conditioning. To improve computational efficiency of large scale three-dimensional computations, the DG-FTLE algorithm is adapted to parallel architectures. Three test cases assess the performance of DG-FTLE on curved geometries and in three-dimensions including the two- and three-dimensional viscous flow over a NACA 65-(1)412 airfoil and the ABC flow. The test cases show the efficacy of the exponential filter and $h$-adaptivity in removing Gibbs oscillations and improving conditioning. The cases also show the ability of DG-FTLE to visualize FTLE fields in three-dimensional, unsteady flows over complex geometries.