TY - JOUR
T1 - Crank-Nicolson Galerkin Approximations for Logarithmic Klein-Gordon Equation
AU - Chen , Fang
AU - Li , Meng
AU - Zhao , Yanmin
JO - Journal of Computational Mathematics
VL - 3
SP - 641
EP - 672
PY - 2024
DA - 2024/11
SN - 43
DO - http://doi.org/10.4208/jcm.2312-m2023-0185
UR - https://global-sci.org/intro/article_detail/jcm/23553.html
KW - Logarithmic Klein-Gordon equation, Finite element method, Cut-off, Error
splitting technique, Convergence.
AB - <p style="text-align: justify;">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 element
method (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 the
three regularized models. Then, the cut-off function technique and the time-space error
splitting technique are innovatively combined to rigorously analyze the unconditionally optimal and high-accuracy convergence results of the numerical scheme without any coupling
condition between the temporal step size and the spatial mesh width. The theoretical
framework is uniform for the other two regularized models. Finally, numerical experiments
are provided to verify our theoretical results. The analytical techniques in this work are
not limited in the FEM, and can be directly extended into other numerical methods. More
importantly, this work closes the gap for the unconditional error/stability analysis of the
numerical methods for the logarithmic systems in higher dimensional spaces.</p>