Energy-preserving RERK-FEM for the regularized logarithmic Schrödinger equation

A high-order implicit–explicit (IMEX) finite element method with energy conservation is constructed to solve the regularized logarithmic Schrödinger equation (RLogSE) with a periodic boundary condition. The discrete scheme consists of the relaxation-extrapolated Runge–Kutta (RERK) method in the temporal direction and the finite element method in the spatial direction. Choosing a proper relaxation parameter for the RERK method is the key technique for energy conservation. The optimal error estimates in the $L^2$-norm and $H^1$-norm are provided without any restrictions between time step size τ and mesh size h by temporal–spatial splitting technology. Numerical examples are given to demonstrate the theoretical results.