An innovative low-order H1-Galerkin mixed finite element framework for superconvergence analysis in nonlinear Klein–Gordon equations

A $H^1$-Galerkin mixed finite element method (MFEM) is explored for solving the nonlinear Klein–Gordon equation, utilizing the lower-order bilinear element paired with the zero-order Raviart–Thomas element $(Q_{11}+Q_{10}\times Q_{01})$. The existence and uniqueness of the solutions for the discretized system are rigorously established. By exploiting the integral identity associated with the bilinear element, a superconvergence estimate linking the interpolation and the Riesz projection is derived. In the semi-discrete framework, the supercloseness of order $O\left(h^2\right)$ is achieved for the primary variable $u$ in the $H^1$ norm and for the auxiliary variable $\overrightarrow{p}=\nabla u$ in the $H(\text{div};\Omega )$ norm, respectively. Additionally, global superconvergence results are obtained through an interpolation-based postprocessing method. Moving to the fully discrete formulation, a second-order scheme is introduced, which exhibits the supercloseness of order $O(h^2+{\tau }^2)$, where $h$ represents the subdivision parameter and $\tau$ the time increment. Finally, the numerical experiments are conducted to confirm the validity of the theoretical findings.