Discontinuous Galerkin finite element method; Accuracy; Stability; Nonlinear Hamilton–Jacobi equations

This paper deals with the formulation, implementation, and application of potentially powerful temporal finite element approximations for solving initial value problems. Weighted residual methods using continuous and discontinuous approximations are developed and their characteristics in terms of accuracy and stability are presented and discussed. An adaptive time spectral element methodology is proposed for Lagrangian spectral elements and its computational performance is demonstrated for stiff systems of ordinary differential equations.