Robust Adaptive Numerical Scheme for Klein-Gordon Equation Based on Exponentially Fitted Splines

The motivation of the current study is to develop a powerful and efficient adaptive numerical scheme for computing time-dependent Klein-Gordon equation using the finite element method. The Klein-Gordon problem in one spatial dimension represents a relativistic wave equation that gives a description of relativistic electrons. It has been observed that the considered Klein-Gordon problem possesses nonlinearity, therefore, quasilinearization technique has been considered to tackle with non-linear behavior arising in the equation. Temporal semi-discretization has been carried out by employing Crank-Nicolson scheme. Furthermore, exponentially fitted splines on a layer-adapted Shishkin mesh have been considered to perform spatial discretization using the finite element method. The stability of the developed scheme has been proved. In the last, numerical experiments have been carried out which shows that the developed adaptive scheme is highly efficient in capturing boundary layers sharply occurring in the findings as becomes small. To the best of the author’s knowledge, no such work is presented till date for small values of .