Unconditional stability and convergence of Crank–Nicolson Galerkin FEMs for a nonlinear Schrödinger–Helmholtz system

Abstract

The paper is concerned with the unconditional stability and optimal $$L^2$$ error estimates of linearized Crank–Nicolson Galerkin FEMs for a nonlinear Schrodinger–Helmholtz system in $${\mathbb {R}}^d$$ ( $$d=2,3$$ ). By introducing a corresponding time-discrete system, we separate the error into two parts, i.e., the temporal error and the spatial error. Since the latter is $$\tau $$ -independent, the uniform boundedness of numerical solutions in $$L^{\infty }$$ -norm follows an inverse inequality immediately without any restrictions on time stepsize. Then, optimal error estimates are obtained in a routine way. Numerical examples in both two and three dimensional spaces are given to illustrate our theoretical results.