Nonconforming quadrilateral finite element method for Ginzburg–Landau equation

Abstract

Nonconforming finite element method is studied for a linearized backward fully-discrete scheme of the Ginzburg-Landau equation with the quadrilateral EQ rot 1 element. The unconditional convergent result of order O(h + ) in the broken H 1 -norm is deduced rigorously based on a splitting technique, by which the ratio between the subdivision parameter h and the time step is removed. Furthermore, numerical results are provided to confirm the theoretical analysis. The analysis developed herein can be regarded as a framework to deal with the unconditional convergent analysis of the Ginzburg-Landau equation for other known low order nonconforming elements. KEYWORDS EQ rot1 element, Galerkin method, Ginzburg-Landau equation, quadrilateral EQ rot 1 element, linearized scheme, quadrilateral meshes, unconditional convergent estimates INTRODUCTIONThe Ginzburg-Landau equation is an important nonlinear evolution equation, which can be used to describe a vast of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity and Bose-Einstein condensation to liquid crystal and strings in field theory [1]. The existence and uniqueness of weak solution, the regularity and long-time behavior of the solution for the problem can be found in [2][3][4][5]. Numerical methods for Ginzburg-Landau equation include spectral method [6][7][8], finite difference method [9,10], and finite element method (FEM) [11,12] have already been studied. In this article, we are concerned with the following complex Ginzburg-Landau equationu(X, t) = 0, (X, t) ∈ Ω × [0, T], u(X, 0) = u 0 (X), X ∈ Ω,(1.1) Numer Methods Partial Differential Eq. 2020;36:329-341. wileyonlinelibrary.com/journal/num

The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation

“…Besides, they applied the modified multiple scales method for the same aims. Recently, the study utilizes nonlinear partial derivatives that have befitted one of the attentive applied mathematics branches [1, 11, 20–23, 26, 36]. Despite the study being done right now, the use of integrals and derivatives has been rare until now.…”

“…One of the sets of differential equations that find applications in real life is the oscillatory problems. In particular, the third-order differential equations arise in many important numbers of physical problems, such as the deflection of a curved beam having a constant or a varying cross-section, three-layer beam, the motion of the rocket, thin-film flow, electromagnetic waves, or gravity-driven flows [1,4,11,16,23,33,36]. Therefore, third-order differential equations have attracted considerable attention over the last three decades, and so many theoretical and numerical studies dealing with such equations have appeared in the literature (see [3,38] and references therein).…”