Dual System Least-Squares Finite Element Method for a Hyperbolic Problem

This study investigates the dual system least-squares finite element method, namely the LL* method, for a hyperbolic problem. It mainly considers nonlinear hyperbolic conservation laws and proposes a combination of the LL* method and Newton's iterative method. In addition, the inclusion of a stabilizing term in the discrete LL* minimization problem is proposed, which has not been investigated previously. The proposed approach is validated using the one-dimensional Burgers equation, and the numerical results show that this approach is effective in capturing shocks and provides approximations with reduced oscillations in the presence of shocks.