Optimal Error Estimates of a Linearized Crank-Nicolson Galerkin FEM for the Kuramoto-Tsuzuki Equations (Communications in Computational Physics, 2019)
This paper is concerned with unconditionally optimal error estimate of thelinearized Galerkin finite element method for solving the two-dimensional and three-dimensional Kuramoto-Tsuzuki equations, while the classical analysis for these nonlinear problems always requires certain time-step restrictions dependent on the spatialmesh size. The key to our analysis is to obtain the boundedness of the numerical approximation in the maximum norm, by using error estimates in certain norms in thedifferent time level, the corresponding Sobolev embedding theorem, and the inverseinequality. Numerical examples in both 2D and 3D nonlinear problems are given toconfirm our theoretical results.
Keywords: Unconditionally optimal error estimates; linearized Galerkin finite element method; Kuramoto-Tsuzuki equation; high-dimensional nonlinear problems.