Error Estimates of a Space‐Time Spectral Method for Nonlinear Klein–Gordon Equation With Unknown Coefficients

Abstract

ABSTRACTA space‐time spectral method combined with mollification method is proposed for the inverse coefficient problem of the nonlinear Klein–Gordon equation. The spectral scheme is utilized to reconstruct an unknown time‐dependent coefficient and wave displacement in a nonlinear Klein–Gordon equation. We apply the Legendre–Galerkin method in spatial direction and the Legendre–Petrov–Galerkin method in temporal direction. We calculate the nonlinear term with the pseudospectral treatment by using Chebyshev‐Gauss‐Lobatto interpolation, which is efficiently computed via the fast Legendre transform. For the perturbed measurements, we apply the appropriate mollification method to obtain stable numerical differentiation and smooth boundary data. Using rigorous error estimates, we establish the convergence and stability of the iterative solution for the fully‐discrete algorithm. Especially, we also present, for the first time, the convergence and stability analysis of the iterative solution that combines spectral methods with regularization techniques. Numerical results show the efficiency and stability of this approach and agree well with the theoretical analysis.