A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Abstract

Abstract. We provide a posteriori error estimates in the L ∞ ([0, T ]; L 2 (Ω))−norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schrödinger equations up to the critical exponent. In particular for the discretization in time we use the relaxation Crank-Nicolson-type scheme introduced by Besse in [9]. The space discretization consists of finite element spaces that are allowed to change between time steps. The estimates are obtained using the reconstruction technique. Through this technique the problem is converted to a perturbation of the original partial differential equation and this makes it possible to use nonlinear stability arguments as in the continuous problem. Our analysis includes as special cases the cubic and quintic nonlinear Schrödinger equations in one spatial dimension and the cubic nonlinear Schrödinger equation in two spatial dimensions. Numerical results illustrate that the estimates are indeed of optimal order of convergence.

On the reflection of solitons of the cubic nonlinear Schrödinger equation

“…Formally, we expect the method to be second order accurate in time, and of r + 1‐order accurate in space, which can be expressed by an a priori error estimate of the following form: where C is a constant depending on the exact solution u of and data of the problem, but it is independent of h and k . A similar estimate was proven rigorously in and in the form of a posteriori error bound in . In , an analogous a priori error estimate was obtained using finite differences.…”

“…where C is a constant depending on the exact solution u of (1) and data of the problem, but it is independent of h and k. A similar estimate was proven rigorously in [7] and in the form of a posteriori error bound in [9]. In [3], an analogous a priori error estimate was obtained using finite differences.…”

“…However, the relaxation scheme does not preserve the energy E (·) in two space dimensions. The relaxation Crank–Nicolson method is also used in where optimal a posteriori error estimates for models of type are obtained. Based on these a posteriori error estimates, one can derive a space‐time adaptive algorithm, which will be able to capture all interested features of the solution with substantial reduction of the overall computational cost compared with that of uniform grids.…”

Error estimation of the Besse Relaxation Scheme for a semilinear heat equation

“…5). It is worth to note that, in the bibliography, the Relaxation Scheme is formulated along with the initial choice Φ 1 2 := ( 0 ) (see, e.g., [4,5,10,11]), which is a first order in time approximation of ( ( 1 2 , •)) and results a first order in time convergence of Φ + 1 2 to ( ( + 1 2 , •)) (see Thm. 4.7 and Tab.…”

“…Oelz and Trabelsi [14] formulated a time-discrete version of the (RS) for the approximation of the solution to the Cauchy problem for a special nonlinear Schrödinger equation occurring in plasma physics, and then developed a convergence analysis analogous to that of [5], without, also, arriving at a conclusion on the order of convergence. Katsaounis and Kyza [10] first proposed a finite element version of the (RS) over a non uniform partition of the time interval, and then constructed an posteriori bound for the error only at the time-nodes, under the assumption that the proposed method has a second order convergence at the intermediate time nodes. At this point, we would like to observe, that the finite element version of the (RS) proposed in [10,11] requires the solution of two linear systems of algebraic equations at every time-step, and thus its computational complexity is two times higher than that of the corresponding finite difference version of the (RS).…”