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

Applied Mathematics and Computation 367 (2020) 124772

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Uniformly superconvergent analysis of an efficient two-grid method for nonlinear Bi-wave singular perturbation problem Dongyang Shi∗, Yanmi Wu School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001 China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 17 April 2019 Revised 29 July 2019 Accepted 22 September 2019

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 L2 − norm are obtained, which are independent of the perturbation parameter δ . Some numerical results indicate that the proposed method is indeed an efficient algorithm.

Keywords: Nonlinear Bi-wave singular perturbation problem Two-grid method Ciarlet–Raviart scheme Existence and uniqueness Uniformly superconvergent estimates

© 2019 Elsevier Inc. All rights reserved.

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

⎧ 2 ⎨δθ φ − φ + f (φ ) = g(x, y ),

in ,

⎩φ = ∂φ = 0, ∂ n¯

on

(1)

∂ .

Where θ is the wave operator,

θφ =

∂ 2φ ∂ 2φ − , ∂ x2 ∂ y2

θ 2φ =

∂ 4φ ∂ 4φ ∂ 4φ −2 2 2 + , ∂ x4 ∂x y ∂ y4

n¯ = (n1 , −n2 ),

∂φ = ∇φ · n¯ , ∂ n¯

 ⊂ R2 is a bounded domain with the boundary ∂ , and n = (n1 , n2 ) denotes the unit outward normal to ∂ , 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 (φ ) = 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 ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (D. Shi).

https://doi.org/10.1016/j.amc.2019.124772 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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D. Shi and Y. Wu / Applied Mathematics and Computation 367 (2020) 124772

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 (φ ) = f (x, y ) and f(φ ) is a nonlinear monotonically increasing function of Type(I ) : f (φ ) = κ1 φ m (κ1 > 0, and m is a positive odd number); or Type(II ) : f (φ ) = κ2 eφ (κ2 > 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 EQ1rot 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 ψ = δθ φ in L2 − 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 () with the norm ·, and let H01 () = {v ∈ H 1 () : v|∂  = 0}. Further, we quote the classical Sobolev spaces Wm,p ()(1 ≤ p ≤ ∞) with norm ||·||m,p . We simply write ||·||m,p as ||·||m while p = 2. 2. Preliminaries Assume that h is a regular rectangular subdivision of  with mesh size h ∈ (0, 1). For a given e ∈ h , we denote its four vertices and edges are Ai and Li = Ai Ai+1 (i = 1, 2, 3, 4.mod (4 )), respectively. Then we define the EQ1rot element space Wh as [15]:

Wh = {wh : wh |e ∈ span{1, x, y, x2 , y2 },



L

[wh ]ds = 0, L ⊂ ∂ e, ∀ e ∈ h },

where [wh ] represents the jump of wh across the internal edge L, and it is wh itself if L belongs to ∂ . The associated interpolation operator Ih |e = Ie over Wh is defined by:





Li

(Ie ϕ − ϕ )ds = 0, i = 1 ∼ 4, (Ie ϕ − ϕ )dxdy = 0. e

Then, for wh ∈ Wh , ϕ the following results, which are helpful to our uniform error estimate analysis, have been proven in [16,17] and [18,19], respectively. ∈ H4 (),



 ∂ (ϕ − Ih ϕ ) ∂ wh dxdy = ∂x ∂x e



∂ (ϕ − Ih ϕ ) ∂ wh dxdy = 0, ∂y ∂y e e e    ∂ϕ   ∂ϕ O(h2 )ϕ3 wh h , (ii ) nx wh ds = ny wh ds = O(h2 )ϕ4 wh 0 , ∂ x ∂ y ∂e ∂e e e (i )

(iii ) wh 0,2m ≤ C wh h , Here and later,  · h = ( irrelevant to h and δ .

 e

m = 1, 2, . . . .

(2)

(3) (4)

1 2

| · |21,e ) is a norm on Wh , C > 0 (with or without subscript) denotes a general constant, which is

3. Existence and uniqueness of the discrete problem Now, introducing ψ =

√

δθ φ , we can rewrite problem (1) as the system:

⎧ √ ψ = δθ φ , in , ⎪ ⎪ ⎪ ⎨√ δθ ψ − φ + f (φ ) = g(x, y ), ⎪ ⎪ ⎪ ⎩φ = ∂φ = 0, on ∂ . ∂ n¯

in ,

(5)

We pose the weak formulation of (5) to find {ψ , φ} ∈ H 1 () × H01 (), such that



√

(ψ , ω ) + δ (∇¯ φ , ∇ω ) = 0, ∀ ω ∈ H 1 (), √

(∇ φ , ∇ χ ) − δ (∇¯ ψ , ∇χ ) + ( f (φ ), χ ) = (g, χ ), ∀ χ ∈ H01 (), ¯ v = ( ∂v , − ∂v ). where ∇ ∂x ∂y

(6)

D. Shi and Y. Wu / Applied Mathematics and Computation 367 (2020) 124772

3

The existence and uniqueness of the solution of (6) can be achieved easily by proving the equivalence between the problem (1) or (5) and the weak form (6) with the similar arguments as [20,21]. The discrete approximation to (6) is seek {ψ h , φ h } ∈ Wh × Wh , such that



√

(ψh , ωh ) + δ (∇¯ h φh , ∇h ωh )h = 0, ∀ ωh ∈ Wh , √

(∇h φh , ∇h χh )h − δ (∇¯ h ψh , ∇h χh )h + ( f (φh ), χh ) = (g, χh ), ∀ χh ∈ Wh ,

(7)

¯ denotes the gradient operators piecewisely, (∇ ˆ ∗, ∇ ˆ  ) =  (∇ ˆ ∗, ∇ ˆ  )e , (∇ ˆ = ∇ or ∇ ¯ ). where ∇ h or ∇ h h h h e

Now we will turn to analyze the solvability of the discrete system (7) through the Brouwer fixed point theorem inspired by the ideas of [22–24]. Theorem 3.1. For any constant 0 < δ ≤ 1, problem (7) has a unique solution (φ h , ψ h ). In particular, for Type (I), we have

ψh 0 + φh h ≤ C g0 .

(8)

Proof. We start to prove the existence of the solution (ψ h , φ h ) of problem (7). In fact, for (ωh , χ h ) ∈ Wh × Wh , from (6) and (7), we get

⎧ √ √  ¯ h ( φ − φh ) , ∇ h ω h ) h = δ ⎪ ( ψ − ψh , ωh ) + δ ( ∇ ∇¯ φ · n ωh ds, ⎪ ⎪ ∂e ⎪ e ⎪ ⎨ √ (∇h (φ − φh ), ∇h χh )h − δ (∇¯ h (ψ − ψh ), ∇h χh )h + ( f (φ ) − f (φh ), χh ) ⎪ ⎪ ⎪  √  ⎪ ⎪ ∇φ · n χh ds − δ ∇¯ ψ · n χh ds. ⎩= e

∂e

e

(9)

∂e

By Taylor’s expansion, we have

1

f (φh ) = f (φ ) + f (φ )(φh − φ ) + f (ζ )(φ − φh )2 , 2 Setting R f (φ − φh ) =

1 2

ζ = φ + σ ( φh − φ ) , 0 ≤ σ ≤ 1 .



f (ζ )(φ − φh )2 , for (ωh , χ h ) ∈ Wh × Wh , we can rewrite (9) as:

⎧ √ √  ¯ h ( φ − φh ) , ∇ h ω h ) h = δ ⎪ ( ψ − ψh , ωh ) + δ ( ∇ ∇¯ φ · n ωh ds, ⎪ ⎪ ∂e ⎪ e ⎪ ⎨ √ (∇h (φ − φh ), ∇h χh )h − δ (∇¯ h (ψ − ψh ), ∇h χh )h + ( f (φ )(φ − φh ), χh ) ⎪ ⎪ ⎪ √   ⎪ ⎪ ∇φ · n χh ds − δ ∇¯ ψ · n χh ds + (R f (φ − φh ), χh ). ⎩= e

∂e

e

(10)

∂e

In order to investigate the above system, we consider the following auxiliary problem: find (ψ˜ h , φ˜ h ) ∈ Wh × Wh , such that

⎧ √ √  ¯ h (φ − φ˜ h ), ∇h ωh )h = δ ⎪ ( ψ − ψ˜ h , ωh ) + δ (∇ ∇¯ φ · n ωh ds, ∀ ωh ∈ Wh , ⎪ ⎪ ∂ e ⎪ e ⎪ ⎨ √ (∇h (φ − φ˜ h ), ∇h χh )h − δ (∇¯ h (ψ − ψ˜ h ), ∇h χh )h + ( f (φ )(φ − φ˜ h ), χh ) ⎪ ⎪ ⎪  √  ⎪ ⎪ ∇φ · n χh ds − δ ∇¯ ψ · n χh ds, ∀ χh ∈ Wh . ⎩= e

Then, when φ that

∂e

∈ H4 (), ψ

e

∈ H3 (),

(11)

∂e

by the repeated argument as [6] and using the properties of (2) and (3), we can prove

√

ψ − ψ˜ h 0 + φ − φ˜ h h ≤ C (1 + δ )h2 .

(12)

For simplicity, we define the norm as :

(ωh , χh )+ = ωh 0 + χh h , ∀ (ωh , χh ) ∈ Wh × Wh . For given (α h , β h ) ∈ Wh × Wh , define the map Sh : Wh × Wh → Wh × Wh by Sh (αh , βh ) = (ψl , φl ) satisfying

⎧ √ √  ¯ ⎪ ( ψ − ψ , ω ) + δ ( ∇ ( φ − φ ) , ∇ ω ) = δ ∇¯ φ · n ωh ds, ∀ ωh ∈ Wh , ⎪ l h h l h h h ⎪ ∂e ⎪ e ⎪ ⎨ √ (∇h (φ − φl ), ∇h χh )h − δ (∇¯ h (ψ − ψl ), ∇h χh )h + ( f (φ )(φ − φl ), χh ) ⎪ ⎪ ⎪ √   ⎪ ⎪ ∇φ · n χh ds − δ ∇¯ ψ · n χh ds + (R f (φ − βh ), χh ), ∀ χh ∈ Wh . ⎩= e

∂e

e

∂e

We set φ − φl = (φ − φ˜ h ) + (φ˜ h − φl ) := ηφ + ξφ , ψ − ψl = (ψ − ψ˜ h ) + (ψ˜ h − ψl ) := ηψ + ξψ .

(13)

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D. Shi and Y. Wu / Applied Mathematics and Computation 367 (2020) 124772

Then by use of (11), we can rewrite (13) as :

⎧ √ √  ¯ h ξφ , ∇ h ω h ) h = δ ⎪ ( ξ , ωh ) + δ ( ∇ ∇¯ φ · n ωh ds, ∀ ωh ∈ Wh , ⎪ ψ ⎪ ∂e ⎪ e ⎪ ⎨ √ (∇h ξφ , ∇h χh )h − δ (∇¯ h ξψ , ∇h χh )h + ( f (φ )ξφ , χh ) ⎪ ⎪ ⎪  √  ⎪ ⎪ ∇φ · n χh ds − δ ∇¯ ψ · n ψh ds + (R f (φ − βh ), χh ), ∀ χh ∈ Wh . ⎩= ∂e

e

(14)

∂e

e

In order to prove the existence of the discrete solution to problem (7), we need to show that the map Sh has a fixed point, i.e., Sh maps a ball Oγ1 (ψ˜ h , φ˜ h ) to itself, where

Oγ1 (ψ˜ h , φ˜ h ) = {(αh , βh ) ∈ Wh × Wh : (αh , βh ) − (ψ˜ h , φ˜ h )+ ≤ γ1 }.

(15)

During our subsequent analysis, it is necessary to prove that the nonlinear term R f (φ − βh ) is bounded. In fact, according to (12) and (15), for (αh , βh ) ∈ Oγ1 (ψ˜ h , φ˜ h ), we have

√

φ − βh 0,4 ≤ C φ − βh h ≤ C φ˜ h − βh h + C φ˜ h − φh ≤ C ((1 + δ )h2 + γ1 ).

(16)

By virtue of (4), (12) and (16), for any χ h ∈ Wh , we get

1 2



|(R f (φ − βh ), χh )| ≤  f (φ + λ(βh − φ ))0,4 φ − βh 20,4 χh 0,4 ≤ C ((1 +

√

√

δ )h2 + γ1 )(h2 + γ1 )2 χh h ≤ C ((1 + δ )h2 + γ1 )γ12 χh h ,

which shows

√

|(R f (φ − βh ), χh )| ≤ C ((1 + δ )h2 + γ1 )γ12 χh h .

(17)

Since the estimates of ηψ and ηφ are received from (12), we now deduce the estimates of ξ ψ and ξ φ so as to get the estimates of ψ − ψl and φ − φl . On one hand, taking ωh = ξψ ∈ Wh and χh = ξφ ∈ Wh in (14) yields

√ 

( ξψ , ξψ ) + ( ∇ h ξφ , ∇ h ξφ ) h + ( f ( φ ) ξ φ , ξφ ) = δ +

 e

e

√ 

∂e

∇φ · n ξφ ds − δ

e

∂e

Applying (3) and (17), we have

∇¯ φ · n ξψ ds

∂e

∇¯ ψ · n ξφ ds + (R f (φ − βh ), ξφ ). √

ξψ 20 + ξφ 2h + ( f (φ )ξφ , ξφ ) ≤ C δ h2 |φ|4 ξψ 0 + Ch2 |φ|3 ξφ h +C

√

√

δ h2 |ψ|3 ξφ h + C ((1 + δ )h2 + γ1 )γ12 ξφ h .

(18)

On the other hand, noting that f (φ ) = κ1 mφ m−1 ≥ 0 (κ 1 > 0, and m is a positive odd number) or f (φ ) = κ2 eφ > 0 (κ 2 > 0), we have

( f ( φ ) ξφ , ξφ ) ≥ 0 , together with (18) and Young’s inequality, we get

√

ξψ 0 + ξφ h ≤ C0 (ψ , φ )((1 + δ )h2 + γ1 )γ12 , which follows that

√

(αh , βh ) − (ψ˜ h , φ˜ h )+ ≤ (ψl , φl ) − (ψ˜ h , φ˜ h )+ = ξψ 0 + ξφ h ≤ C0 (ψ , φ )((1 + δ )h2 + γ1 )γ12 , where C0 (ψ , φ ) is also a positive constant independent of h and δ , but depends on φ and ψ . Thus, let h ≤ (C0 (ψ , φ ))−2 , then the choice γ1 = C0 (ψ , φ )h leads to

(αh , βh ) − (ψ˜ h , φ˜ h )+ ≤ γ1 . That is, for sufficiently small mesh size h, Sh maps the ball of radius γ1 = O(h ) > 0, centered at (ψ˜ h , φ˜ h ) into itself. Now, we derive the uniqueness of (ψ h , φ h ) of (7) by proving that Sh is a contraction mapping in the ball Oγ1 (ψ˜ h , φ˜ h ). In fact, let (ψ 1 , φ 1 ) and (ψ 2 , φ 2 ) be two different solutions of (13), according to (12) and (18), we have

√

ψi − ψ˜ h 0 + φi − φ˜ h h ≤ C˜0 ((1 + δ )h2 + γ1 )γ12 , where C˜0 = max{C0 (ψ1 , φ1 ), C0 (ψ2 , φ2 )}.

i = 1, 2,

D. Shi and Y. Wu / Applied Mathematics and Computation 367 (2020) 124772

5

For any (ωh , χ h ) ∈ Wh × Wh and (α1 , β1 ), (α2 , β2 ) ∈ Oγ1 (ψ˜ h , φ˜ h ), we obtain by (13) that

⎧ √ √  ¯ ⎪ ( ψ − ψ , ω ) + δ ( ∇ ( φ − φ ) , ∇ ω ) = δ ∇¯ φ · n ωh ds, 1 2 1 2 ⎪ h h h h h ⎪ ∂e ⎪ e ⎪ ⎨ √ (∇h (φ1 − φ2 ), ∇h χh )h − δ (∇¯ h (ψ1 − ψ2 ), ∇h χh )h + ( f (φ )(φ1 − φ2 ), χh ) ⎪ ⎪ ⎪ √   ⎪ ⎪ ∇φ · n χh ds − δ ∇¯ ψ · n χh ds + (R f (φ − β1 ) − R f (φ − β2 ), χh ). ⎩= ∂e

e

e

∂e

With the similar calculations as the above, we have

√

ψ1 − ψ2 0 + φ1 − φ2 h ≤ C˜0 ((1 + δ )h2 + γ1 )γ12 (α1 , β1 ) − (α2 , β2 )+ . Inserting h ≤ (C˜0

)−2

(19)

and γ1 = C˜0 h to (19) yields

√

ψ1 − ψ2 0 + φ1 − φ2 h ≤ ((1 + δ )h 2 + 1 )h(α1 , β1 ) − (α2 , β2 )+ , which deduces

√

3

Sh (α1 , β1 ) − Sh (α2 , β2 )+ ≤ ((1 + δ )h 2 + 1 )h(α1 , β1 ) − (α2 , β2 )+ . 3

(20)

Therefore, for sufficiently small h, Sh is a contraction mapping in the ball Oγ1 (ψ˜ h , φ˜ h ). As to the priori estimate of (8) for Type(I ), it can be gained on repeating the argument as [8]. The proof is complete.



4. Two-grid method and uniformly superconvergent analysis In this section, we will present the procedure of a two-grid method for problem (1) by the Ciarlet–Raviart scheme and analyze the corresponding uniformly superconvergent estimates. For this purpose, we define another EQ1rot approximation space WH ⊂ Wh (h  H  1) on the coarse grid H of . Then we establish the two-grid algorithm as follows. Step 1: For {ωH , χ H } ∈ WH × WH , we solve {ψ H , φ H } ∈ WH × WH on the coarse grid H for the following nonlinear system defined by



√

(ψH , ωH ) + δ (∇¯ H φH , ∇H ωH )H = 0,

(21)

√

(∇H φH , ∇H χH )H − δ (∇¯ H ψH , ∇H χH )H + ( f (φH ), χH ) = (g, χH ).

Step 2: For {ωh , χ h } ∈ Wh × Wh , we compute { h , h } ∈ Wh × Wh on the fine grid h for the following linear system satisfying



√

(h , ωh ) + δ (∇¯ h h , ∇h ωh )h = 0, √

(∇h h , ∇h χh )h − δ (∇¯ h h , ∇h χh )h + ( f (φH ) + f (φH )(h − φH ), χh ) = (g, χh ).

(22)

Now we are ready to give the following uniformly superclose estimates of the above TGM. Theorem 4.1. Let {ψ , φ }, {ψ H , φ H } and { h , h } be the solutions of (6), (21) and (22), respectively. Assume that ψ ∈ H3 (), φ ∈ H4 (), there hold

IH ψ − ψH 0 + IH φ − φH H ≤ C1 H 2 , Ih ψ − h 0 + Ih φ − h h ≤ C2 (h2 + H 4 ), where C1 ≥ C (ψ 2 +

√

√

(23)

√

√

(24)

δψ 3 + φ3 + δφ4 ) and C2 ≥ max{C (ψ 2 + δψ 3 + φ3 + δφ4 ), C (φ20,4 + C12 )}.

Proof. The detailed proof of (23) can be found in [8], so we only need to prove (24). In fact, by Taylor’s expansion, we have

1

f (φ ) = f (φH ) + f (φH )(φ − φH ) + f (φH + ν (φ − φH ))(φ − φH )2 , (0 ≤ ν ≤ 1 ). 2

(25)

Let φ − h = (φ − Ih φ ) + (Ih φ − h ) := η + ξ , ψ − h = (ψ − Ih ψ ) + (Ih ψ − h ) := ρ + ϑ . Then for {ωh , χ h } ∈ Wh × Wh , the error equations are derived as follows :

⎧ √ √  √ ⎪ ¯ h ξ , ∇h ωh )h = − ( ρ , ωh ) − δ ( ∇ ¯ h η , ∇h ωh )h + δ ( ϑ , ω ) + δ ( ∇ ∇¯ φ · n ωh ds, ⎪ h ⎪ ⎪ ∂e ⎪ e ⎪ ⎨ √ √  (∇h ξ , ∇h χh )h − δ (∇¯ h ϑ , ∇h χh )h = −(∇h η, ∇h χh )h + δ (∇¯ h ρ , ∇h χh )h + ∇φ · n χh ds ∂e ⎪ e ⎪ 1 ⎪ ⎪ √  ¯

⎪ ⎪ ∇ ψ · n χh ds − ( f (φH )(φ − h ), χh ) − f (φH + ν (φ − φH ))(φ − φH )2 , χh . ⎩− δ 2 e

∂e

(26)

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D. Shi and Y. Wu / Applied Mathematics and Computation 367 (2020) 124772

Choosing ωh = ϑ ∈ Wh and χh = ξ ∈ Wh in (26), we obtain

√

√

(ϑ , ϑ ) + (∇h ξ , ∇h ξ )h = −(ρ , ϑ ) − δ (∇¯ h η, ∇h ϑ )h − (∇h η, ∇h ξ )h + δ (∇¯ h ρ , ∇h ξ )h +

√ 

δ

e

∂e

∇¯ φ · n ϑ ds +

 e

− ( f (φH )(φ − h ), ξ ) −

1 2

√ 

∇φ · n ξ ds − δ

∂e

e

∂e



∇¯ ψ · n ξ ds

f (φH + ν (φ − φH ))(φ − φH )2 , ξ





9 

Ai .

(27)

i=1

In view of (2)-(3), by Young’s inequality, the terms on the right side of (27) can be measured by

A1 = −(ρ , ϑ ) ≤ Ch2  p2 ϑ0 ≤ Ch4 ψ22 +

1 ϑ20 , 3

A2 = A3 = A4 = 0, √  √ 1 A5 = δ ∇¯ φ · n ϑ ds ≤ C δ h2 φ4 ϑ0 ≤ C δ h4 φ24 + ϑ20 , 3 ∂ e e  1 A6 = ∇φ · n ξ ds ≤ Ch2 φ3 ξ h ≤ Ch4 φ23 + ξ 2h , 5 ∂e e and

√  A7 = − δ e

∂e

√

1 5

∇¯ ψ · n ξ ds ≤ C δ h2 ψ3 ξ h ≤ C δ h4 ψ23 + ξ 2h ,

respectively. Then we introduce the following novel splitting technique as [8] to handle the major difficult term.

A8 = −( f (φH )(φ − h ), ξ ) = −( f (φH )(φ − Ih φ ), ξ ) − ( f (φH )(Ih φ − h ), ξ ). Noting that f (φ ) = κ1 mφ m−1 ≥ 0 (κ 1 > 0, and m is a positive odd number) or f (φ ) = κ2 eφ > 0 (κ 2 > 0) again, we have

−( f (φH )(Ih φ − h ), Ih φ − h ) ≤ 0. Thus, by the interpolation theory and Young’s inequality, we have

A8 ≤ |( f (φH )(φ − Ih φ ), ξ )| ≤ C φ − Ih φ0 ξ h ≤ Ch2 φ2 ξ h ≤ Ch4 φ22 +

1 ξ 2h . 5

Moreover, with the help of (4), the interpolation theory and (23), we obtain

A9 ≤ C φ − φH 20,4 ξ h ≤ C (φ − IH φ20,4 + IH φ − φH 20,4 )ξ h ≤ C (H 4 φ22,4 + IH φ − φH 2h )ξ h ≤ C (H 4 φ22,4 + C12 H 4 )ξ h 2

≤ C H 8 (φ22,4 + C12 ) +

1 ξ 2h . 5

Combining the above achievements, we get 2

ϑ 20 + ξ 2h ≤ Ch4 (ψ 22 + δψ 23 + φ23 + δφ24 ) + CH 8 (φ22,4 + C12 ) , which leads to

√

√

ϑ 0 + ξ h ≤ Ch2 (ψ 2 + δψ 3 + φ3 + δφ4 ) + CH 4 (φ22,4 + C12 ) ≤ C2 (h2 + H 4 ). The proof is complete.



Moveover, by utilizing (24) and the interpolated post-processing operator I2h used in [16], the following uniformly global superconvergent estimate can be deduced easily:

φ − I2h h h ≤ C2 (h2 + H 4 ). Remark 1. If we substitute the nonconforming EQ1rot element with the nonconforming rectangular quasi-Wilson element [25] or modified quadrilateral quasi-Wilson element [26], then when ϕ ∈ H4 (), wh ∈ Wh , we have

 e

e

 e

 ∂ (ϕ − Ih ϕ ) ∂ wh dxdy = ∂x ∂x e

∂e

 ∂ϕ nx wh ds = ∂x e



∂e



e

∂ (ϕ − Ih ϕ ) ∂ wh dxdy ≤ Ch3 ϕ4 wh h ≤ Ch2 ϕ4 wh 0 , ∂y ∂y

∂ϕ n w ds ≤ Ch3 ϕ4 wh h ≤ Ch2 ϕ4 wh 0 , ∂y y h

which implies that Theorem (4.1) holds true.

D. Shi and Y. Wu / Applied Mathematics and Computation 367 (2020) 124772

7

Table 1 Numerical results of φ for δ = 0.001. H

h

Ih φ − h h

order

I2h h − φh

order

1/2 1/4 1/6 1/8

1/4 1/16 1/36 1/64

1.1700E-01 1.0417E-02 2.2801E-03 7.4228E-04

− 1.7447 1.8735 1.9505

9.5388E-01 7.5898E-02 1.5247E-02 4.8393E-03

− 1.8258 1.9792 1.9946

Table 2 Numerical results of φ for δ = 1.0. H

h

Ih φ − h h

order

I2h h − φh

order

1/2 1/4 1/6 1/8

1/4 1/16 1/36 1/64

9.1561E-02 1.3602E-03 1.3862E-04 3.2174E-05

− 3.0364 2.8161 2.5386

9.2851E-01 7.5113E-02 1.5073E-02 4.7818E-03

− 1.8139 1.9806 1.9954

Table 3 Numerical results of ψ for δ = 0.001 − 1.0. H

h

 ψ − h  0 (δ = 0.001 )

order

 ψ − h  0 ( δ = 1.0 )

order

1/2 1/4 1/6 1/8

1/4 1/16 1/36 1/64

5.8460E-02 2.4404E-03 4.9075E-04 1.5705E-04

− 2.2911 1.9780 1.9803

2.0639E-01 3.7541E-03 4.8239E-04 1.3460E-04

− 2.8904 2.5302 2.2185

Remark 2. When Wh is the conforming quadratic element space, it has been shown in [27] that, for ϕ ∈ H6 (), wh ∈ Wh , there hold

 

 

∂ (ϕ − Ih ϕ ) ∂ wh dxdy = ∂x ∂x

 

∂ (ϕ − Ih ϕ ) ∂ wh dxdy ≤ Ch4 ϕ6 wh 0 , ∂y ∂y

(ϕ − Ih ϕ )wh dxdy ≤ Ch4 ϕ4 wh 0 .

Obviously, this element is also applicable to problem (1) for the proposed two-grid method and the errors are uniformly convergent at a rate of O(h4 ). Remark 3. For the conforming triangular linear element [28], the nonconforming square Q1rot element [29] and the rectangular CNQ1rot element [30], because they don’t satisfy (2) and (3) synchronously, whether (23) and (24) hold or not is an open issue. Therefore, the EQ1rot element used in our present study is a suitable choice. On the other hand, how to extend the number m in the Type(I ) to be a general number also needs to be considered in the future study. 5. Numerical results We consider the following numerical example to verify the theoretical analysis [8] :

⎧ 2 3 ⎨δθ φ − φ + φ = g(x, y ),

in ,

⎩φ = ∂φ = 0, ∂ n¯

on

∂ .

(28)

Where  = [0, 1] × [0, 1], function g(x, y) is computed from the exact solution φ = sin2 (π x )sin2 (π y ). We choose H 4 = h2 in the computation and list the errors in Tables 1–3 for δ = 0.001 ∼ 1.0, respectively. It can be seen that when h → 0, Ih φ − h h , φ − I2h h h and ψ − h 0 are uniformly convergent at a rate of O(h2 ). Meanwhile, we also plot the error reduction results in Fig. 1(a ) − (c ), respectively, and the slope of the curve indicates the convergent rate of this proposed method. All these results are in accordance with our theoretical analysis. 1 On the other hand, we choose the same partition (h = 36 ) to compare the computing cost of the conventional mixed FEM and the two-grid method (TGM) in Table 4 for different δ . It can be seen clearly that the CPU time required for the latter one is only less than 40% of the former one. We also plot the superconvergent errors for φ in Fig. 1(d), which shows that the errors acquired by the TGM have little visual difference from that of the traditional mixed FEM. Therefore, our method is certainly a very efficient algorithm for solving the nonlinear Bi-wave singular perturbation problem (1).

8

D. Shi and Y. Wu / Applied Mathematics and Computation 367 (2020) 124772 0

0

10

10

−1

10 −1

Errors of u for δ=1.0

Errors of u for δ=0.001

10

−2

10

−2

10

−3

10

−3

10

−4

10 Superclose errors of u Superconvergent errors of u

−4

10

0

1

10

Superclose errors of u Superconvergent errors of u

−5

10

2

10 Number of elements

0

1

10

10

2

10 Number of elements

10

0

Superconvegent errors of φ for δ=0.001~1.0

10

Errors of p for δ=0.001~1.0

−1

10

−2

10

−3

10

2

L errors of p for δ=0.001

0.025

0.02

0.015

0.01

0.005

L2 error of p for δ=1.0

−4

10

Superconvegent errors of φ of FEM Superconvegent errors of φ of TGM

0

10

1

2

10 Number of elements

20

10

40

60

80 100 120 Number of elements

140

160

180

Fig. 1. Error results of φ and ψ for different δ . Table 4 Errors and CPU cost of mixed FEM and TGM.

δ

I2h φh − φh

CPU time (s) (mixed FEM)

I2h h − φh

CPU time (s) (TGM)

0.001 0.01 0.1 1.0

1.5289E-02 1.5207E-02 1.5092E-02 1.5073E-02

59.5002 59.3581 59.2442 60.7364

1.5247E-02 1.5182E-02 1.5085E-02 1.5247E-02

22.3423 21.9145 23.4730 23.2635

Acknowledgment This work was supported by the National Natural Science Foundation of China (Nos. 11671369; 11271340). References [1] [2] [3] [4]

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