A new formulation for solving the Eikonal equation is investigated using a time-dependent Hamilton-Jacobi equation. A discontinuous Galerkin (DG) finite element method is proposed for direct reconstruction of the traveltime field as the final stationary solution. Both isotropic and tilted transversely isotropic (TTI) implementations are performed in heterogeneous media with stable and accurate results as long as we honor the high-frequency approximation. We introduce outgoing conditions at edges able to handle complex topography and we deal with singularity at the source through the additive factorization. Expected convergence behavior regarding element interpolation is observed when considering factorization. Comparison between DG and finite difference solutions in the complex BP TTI model with unstructured and structured meshes illustrates the highly accurate traveltime estimation of this DG approach, pointing out perspectives for integrating this accurate local Eikonal solver into efficient methods for getting the stationary solution, such as fast sweeping methods.