Conservative primal hybrid finite element method for weakly damped Klein-Gordon equation

Based on the primal hybrid finite element method (FEM) to discretize spatial variables, a semi-discrete scheme is obtained for the weakly damped Klein-Gordon equation. It is shown that this method is energy-conservative, and optimal error estimates in the energy norm are proved with the help of a modified elliptic projection. Moreover, a superconvergence result is derived, and as a consequence, the maximum norm estimate is obtained. Then, a non-standard type argument shows optimal error analysis in the $L^{\infty }\left(L^2\right)$-norm with reduced regularity assumption on the solution. Further, the optimal order of convergence for the Lagrange multiplier is also established, and a superconvergence result for the gradient of the error between the modified elliptic projection and the primal hybrid finite element solution in maximum norm is derived. For a complete discrete scheme, an energy-conservative finite difference method is applied in the temporal direction, and the well-posedness of the discrete system is shown using a variant of the Brouwer fixed point theorem. The optimal rate of convergence for the primal variable in energy and $L^2$-norm for the fully discrete problem are established. Both semidiscrete and fully discrete schemes are analyzed for polynomial non-linearity, which is of the locally Lipschitz type. Finally, some numerical experiments are conducted to validate our theoretical findings.