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

Abstract 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 ‖ U h n ‖ 0 , ∞ + ‖ ∂ t U h n ‖ 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 ( h 2 + τ 2 ) in H 1 -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.

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