Linearly Implicit Finite Element Methods Approximating the Solution to the Nonlinear Schrödinger Equation with a Schamel-Type Nonlinearity

We consider an initial- and Dirichlet boundary- value problem for a nonlinear Schrödinger equation of the form uₜ=\,i\, u+i\,V\,u+i\,μ \,|u|β\,u+f u t = i Δ u + i V u + i μ | u | β u + f over [0,T]× [ 0 , T ] × Ω , where $$T>0$$ T > 0 , ⊂ Rᵈ Ω ⊂ R d for d∈ \1,2,3\ d ∈ { 1 , 2 , 3 } , β ∈ (0,1) β ∈ ( 0 , 1 ) , V is a real-valued time-independent potential and μ μ is a nonzero real number. The solution to the problem is approximated by the Linearized Backward Euler finite element () method which is dissipative and the Linearized Crank–Nicolson finite element () one which is conservative. Letting τ τ be the time-step and h be the width of the finite element partition of the space domain, we provide an optimal order O(τ +h²) O ( τ + h 2 ) error estimate in the L² L 2 norm for both methods, and an O(τ α+h) O ( τ α + h ) error estimate in the H¹