# Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear Schrödinger Equation

Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear... A linearized Crank–Nicolson Galerkin finite element method with bilinear element for nonlinear Schrödinger equation is studied. By splitting the error into two parts which are called the temporal error and the spatial error, the unconditional superconvergence result is deduced. On one hand, the regularity for a time-discrete system is presented based on the proof of the temporal error. On the other hand, the classical Ritz projection is applied to get the spatial error with order $$O(h^2)$$ O ( h 2 ) in $$L^2$$ L 2 -norm, which plays an important role in getting rid of the restriction of $$\tau$$ τ . Then the superclose estimates of order $$O(h^2+\tau ^2)$$ O ( h 2 + τ 2 ) in $$H^1$$ H 1 -norm is arrived at based on the relationship between the Ritz projection and the interpolated operator. At the same time, global superconvergence property is arrived at by the interpolated postprocessing technique. At last, three numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and $$\tau$$ τ is the time step. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Scientific Computing Springer Journals

# Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear Schrödinger Equation

, Volume 72 (3) – Feb 18, 2017
26 pages

/lp/springer_journal/unconditional-superconvergence-analysis-of-a-crank-nicolson-galerkin-PzHUNNxB81
Publisher
Springer US
Subject
Mathematics; Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Theoretical, Mathematical and Computational Physics
ISSN
0885-7474
eISSN
1573-7691
D.O.I.
10.1007/s10915-017-0390-2
Publisher site
See Article on Publisher Site

### Abstract

A linearized Crank–Nicolson Galerkin finite element method with bilinear element for nonlinear Schrödinger equation is studied. By splitting the error into two parts which are called the temporal error and the spatial error, the unconditional superconvergence result is deduced. On one hand, the regularity for a time-discrete system is presented based on the proof of the temporal error. On the other hand, the classical Ritz projection is applied to get the spatial error with order $$O(h^2)$$ O ( h 2 ) in $$L^2$$ L 2 -norm, which plays an important role in getting rid of the restriction of $$\tau$$ τ . Then the superclose estimates of order $$O(h^2+\tau ^2)$$ O ( h 2 + τ 2 ) in $$H^1$$ H 1 -norm is arrived at based on the relationship between the Ritz projection and the interpolated operator. At the same time, global superconvergence property is arrived at by the interpolated postprocessing technique. At last, three numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and $$\tau$$ τ is the time step.

### Journal

Journal of Scientific ComputingSpringer Journals

Published: Feb 18, 2017

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