Spectral Galerkin method for Cahn-Hilliard equations with time periodic solution

Shimin Chai; Chenguang Zhou

doi:10.3934/dcdsb.2023212

Article Contents

2024, Volume 29, Issue 7: 3046-3057. Doi: 10.3934/dcdsb.2023212

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Spectral Galerkin method for Cahn-Hilliard equations with time periodic solution

Shimin Chai^1, and
Chenguang Zhou^2, ,

1.
School of Mathematics, Jilin University, Changchun 130012, China
2.
Department of Mathematics, Beijing University of Technology, Beijing 100124, China

^*Corresponding author: Chenguang Zhou
^*Corresponding author: Chenguang Zhou

Received: August 2023

Revised: November 2023

Early access: December 25, 2023

Published online: December 25, 2023

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Abstract

This paper is concerned with the numerical approximation to the one-dimensional Cahn-Hilliard equation with time periodic solution. We adopt the implicit Euler method and the spectral Galerkin method for the temporal and spatial discretization, respectively. Then the error estimates are proved for both the semi-discrete and the fully discrete schemes. Numerical experiments are carried out to confirm our theoretical results.

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Mathematics Subject Classification: Primary: 65M70, 65M15.

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Figure 1. The fully discrete approximation solution

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Table 1. The error between the exact solution and the numerical solution

$ k $	$ \mathfrak{E}(k) $	Convergence Order
$ 1 {\rm{e}} $-$ 2 $	$ 7.975 {\rm{e}} $-$ 3 $	–
$ 5 {\rm{e}} $-$ 3 $	$ 4.092 {\rm{e}} $-$ 3 $	0.9629
$ 2.5 {\rm{e}} $-$ 3 $	$ 2.045 {\rm{e}} $-$ 3 $	1.0001
$ 1.25 {\rm{e}} $-$ 3 $	$ 9.973 {\rm{e}} $-$ 4 $	1.0364
$ 6.25 {\rm{e}} $-$ 4 $	$ 4.671 {\rm{e}} $-$ 4 $	1.0943

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