Uniformly superconvergent analysis of an efficient two-grid method for nonlinear Bi-wave singular perturbation problem

Abstract

The main aim of this paper is to present a two-grid method for the fourth order nonlinear Bi-wave singular perturbation problem with low order nonconforming finite element based on the Ciarlet–Raviart scheme. The existence and uniqueness of the approximation solution are demonstrated through the Brouwer fixed point theorem and the uniform superconvergent estimates in the broken $H^1-$ norm and $L^2-$ norm are obtained, which are independent of the perturbation parameter δ. Some numerical results indicate that the proposed method is indeed an efficient algorithm.

Introduction

In this paper, we consider the following fourth order nonlinear Bi-wave singular perturbation problem:

$$\{\begin{array}{cc}\delta {\theta }^2\varphi -\Delta \varphi +f\left(\varphi \right)=g(x,y),\hfill & in\Omega ,\hfill \\ \varphi ={\displaystyle \frac{\partial \varphi }{\partial \overline{n}}}=0,\hfill & on\partial \Omega .\hfill \end{array}$$

Where θ is the wave operator,

$$\theta \varphi ={\displaystyle \frac{{\partial }^2\varphi }{\partial x^2}}-{\displaystyle \frac{{\partial }^2\varphi }{\partial y^2}},{\theta }^2\varphi ={\displaystyle \frac{{\partial }^4\varphi }{\partial x^4}}-2{\displaystyle \frac{{\partial }^4\varphi }{\partial x^2{y}^2}}+{\displaystyle \frac{{\partial }^4\varphi }{\partial y^4}},\overline{n}=(n_1,-n_2),{\displaystyle \frac{\partial \varphi }{\partial \overline{n}}}=\nabla \varphi ·\overline{n},$$

Ω ⊂ R² is a bounded domain with the boundary ∂Ω, and $n=(n_1,n_2)$ denotes the unit outward normal to ∂Ω, 0 < δ ≤ 1 is a real perturbation parameter and problem (1) will turn into the Poisson’ equation when δ tends to zero. g(x, y) and f(ϕ) are known smooth functions (see [1]).

Problem (1) describes the model of the Ginzburg–Landau-type d-wave superconductor (see [2]) and some theoretical and numerical analysis have been concentrated on it. For example, two conforming Galerkin finite element methods (FEMs) and the modified Morley-type discontinuous Galerkin FEMs were analyzed when $f\left(\varphi \right)=f(x,y),g(x,y)=0,$ and optimal order error estimates were acquired in [3] and [4], respectively. But there exist two big defects: one is that the finite element space used in [3] must contain C¹ piecewise polynomials of degree  ≥ 3, which makes the structure quite complicated and the computing cost rather expensive (see [5], [6]). The other is that the estimate for ϕ in the broken $H^1-$ norm in [4] is not uniform with respect to δ. To get rid of the above defects, the appropriate low order conforming and nonconforming mixed FEMs were applied in [7], [8] to dispose of the problem (1) when $f\left(\varphi \right)=f(x,y)$ and f(ϕ) is a nonlinear monotonically increasing function of $\mathbf{Type}\left(I\right):f\left(\varphi \right)={\kappa }_1{\varphi }^m({\kappa }_1>0,$ and m is a positive odd number); or $\mathbf{Type}\left(II\right):f\left(\varphi \right)={\kappa }_2{e}^{\varphi }({\kappa }_2>0$), and uniformly superconvergent estimate results were gained, respectively.

As we know, the two-grid method was put forward in [9] as an efficient algorithm to deal with the nonlinear problems such as the parabolic equation [10], [11], the hyperbolic problem [12], the Navier-Stokes equation [13], the Maxwell equation [14], and so on. However, there is no uniformly convergent estimates or superconvergent results of a two-gird method for problem (1) in the existing literature.

In this article, as an attempt, we will develop a two-grid method with the nonconforming $E{Q}_1^{rot}$ element (see [15]) based on the Ciarlet–Raviart scheme for the problem (1) of the above two types of f(ϕ), and prove the existence and uniqueness of the numerical solution by use of the Brouwer fixed point theorem. Then, the uniformly superconvergent results for ϕ in the broken $H^1-$ norm and uniformly optimal error estimate for $\psi =\sqrt{\delta }\theta \varphi$ in $L^2-$ norm are derived with the help of the typical behaviour of this element. Finally, numerical results are provided to show that the proposed two-grid method can save a lot of computing cost compared with [8] without losing accuracy.

Throughout this paper, we define the natural inner production (·, ·) in L²(Ω) with the norm ‖·‖, and let $H_0^1\left(\Omega \right)=\{v\in H^1\left(\Omega \right):v{|}_{\partial \Omega }=0\}$. Further, we quote the classical Sobolev spaces W^m,p(Ω)(1 ≤ p ≤ ∞) with norm ||·||_m,p. We simply write ||·||_m,p as ||·||_m while $p=2$.