Unconditional superconvergence analysis of a linearized Galerkin FEM for nonlinear hyperbolic equations

Unconditional superconvergence analysis for nonlinear hyperbolic equations with bilinear finite element is studied. A linearized Galerkin finite element method (FEM) is developed and a time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, a rigorous analysis for the regularity of the time-discrete system is presented based on the temporal error estimation skillfully. On the other hand, the numerical solution ‖Uhn‖0,∞+‖∂̃tUhn‖0,∞ is bounded by the spatial error which is deduced with the help of the Ritz projection operator. The superclose and global superconvergence estimates of u with order O(h2+τ2) in H1-norm are derived without any restriction of τ through the relationship between the interpolation operator and the Ritz projection operator. At last, numerical results are provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ, the time step.