Crank-Nicolson Galerkin Approximations for Logarithmic Klein-Gordon Equation (Journal of Computational Mathematics, 2024)
This paper presents three regularized models for the logarithmic Klein-Gordon equation. By using a modified Crank-Nicolson method in time and the Galerkin finite elementmethod (FEM) in space, a fully implicit energy-conservative numerical scheme is constructed for the local energy regularized model that is regarded as the best one among thethree regularized models. Then, the cut-off function technique and the time-space errorsplitting technique are innovatively combined to rigorously analyze the unconditionally optimal and high-accuracy convergence results of the numerical scheme without any couplingcondition between the temporal step size and the spatial mesh width. The theoreticalframework is uniform for the other two regularized models. Finally, numerical experimentsare provided to verify our theoretical results. The analytical techniques in this work arenot limited in the FEM, and can be directly extended into other numerical methods. Moreimportantly, this work closes the gap for the unconditional error/stability analysis of thenumerical methods for the logarithmic systems in higher dimensional spaces.