Convergence Analysis of a Schrodinger Problem with Moving Boundary

TL;DR

The article analyzes convergence for a nonlinear Schrödinger equation on a moving boundary by transforming the problem to a cylindrical domain, yielding advection and time-dependent coefficients. It develops a linearized Crank–Nicolson Galerkin method with semidiscrete and fully discrete formulations, and establishes optimal $L^2$-norm error bounds of $O( au^2 + h^r)$ under the step constraint $\tau = o(h^{n/4})$, alongside $O(h^r)$ in $L^2$ and $O(h^{r-1})$ in $H^1$ for the semidiscrete case. The analysis hinges on a time-dependent Ritz projection, uniform $L^ty$ bounds, and a decomposition-based error framework with Gronwall arguments. Numerical experiments in 1D and 2D confirm the predicted convergence rates for various polynomial bases and both homogeneous and nonhomogeneous problems, illustrating the method’s effectiveness on moving-boundary domains. The results advance understanding of finite-element convergence for nonlinear Schrödinger equations in noncylindrical domains and support applications involving moving interfaces or boundary control.

Abstract

In this article, we present the mathematical analysis of the convergence of the linearized Crank-Nicolson Galerkin method for a nonlinear Schrodinger problem related to a domain with a moving boundary. The convergence analysis of the numerical method is carried out for both semi-discrete and fully discrete problems. An optimal error estimate in the $L^2$-norm with order ${O}(τ^2+ h^s),~ 2\leq s\leq r$, where $h$ is the finite element mesh size parameter, $τ$ is the time step, and $r-1$ represents the degree of the finite element polynomial basis. Numerical simulations are provided to confirm the consistency between theoretical and numerical results, validating the method and the order of convergence for different degrees $p\geq 1$ of the Lagrange polynomials and also for Hermite polynomials (degree $p=3$), which form the basis of the approximate solution.

Figures (7)

• Figure 1: Functions $k(t)$ used to define the noncylindrical domains explored in this work.
• Figure 2: Non-homogeneous 1D problem. Error plot $E(h,\tau)$ with $\tau=h^{\frac{p+1}{2}}$ for boundaries 1, 2 and 3.
• Figure 3: Non-homogeneous 2D problem. Error plot $E(h,\tau)$ with $\tau=h^{\frac{p+1}{2}}$ for boundaries 1, 2 and 3.
• Figure 4: Homogeneous 1D problem. Real part of approximate solutions relative to boundaries 1, 2, and 3, respectively. We consider cubic Hermite basis, $h=1/32$ and $\tau=1/400$.
• Figure 5: Homogeneous 1D problem. Imaginary part of approximate solutions relative to boundaries 1, 2, and 3, respectively. We consider cubic Hermite basis, $h=1/32$ and $\tau=1/400$.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Remark 1
• Remark 2
• Proposition 3
• Remark 3
• Lemma 4
• Remark 4
• Remark 5
• proof