Lower bounds of error estimates for singular perturbation problems with Jacobi-Spectral approximations

Accepted Manuscript Lower bounds of error estimates for singular perturbation eigenvalue problems with Jacobi-Spectral approximations

Juan Zhang, Huanzhen Chen

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S0893-9659(18)30240-4 https://doi.org/10.1016/j.aml.2018.07.019 AML 5591

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Applied Mathematics Letters

Received date : 12 May 2018 Revised date : 12 July 2018 Accepted date : 12 July 2018 Please cite this article as: J. Zhang, H. Chen, Lower bounds of error estimates for singular perturbation eigenvalue problems with Jacobi-Spectral approximations, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.07.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Lower bounds of error estimates for singular perturbation eigenvalue problems with Jacobi-Spectral approximationsI Juan Zhang, Huanzhen Chen∗ School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, P.R. China

Abstract With weighted orthogonal Jacobi polynomials, we study spectral approximations for singular perturbation problems on an interval. The singular parameters of the model are included in the basis functions, and then its stiff matrix is diagonal. Considering the estimations for weighted orthogonal coefficients, a special technique is proposed to investigate the a posteriori error estimates. In view of the difficulty of a posteriori error estimates for spectral approximations, we employ a truncation projection to study lower bounds for the models. Specially, we present the lower bounds of a posteriori error estimates with two different weighted norms in details. Keywords: Generalized Jacobi polynomial, spectral approximation, a posteriori error estimate, singular perturbation eigenvalue problem Contents 1 Introduction

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2 Preliminaries

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3 Lower bound of a posteriori error estimates 3.1 The a posteriori error estimator with L2ω−1,−1 -norm . . . . . . . . . . . . . . . . 3.2 The a posteriori error estimator with Hω−1 1,1 -norm . . . . . . . . . . . . . . . . .

4 4 6

4 Conclusion and remarks

6

1. Introduction Singularly perturbed problems are generally characterized by dynamics operating on multiple scales and very important in practice. Numerical methods for this differential equations have been extensively considered in the scientific and mathematical community. An extensive list of literature on this topic with few different perspectives, containing finite element methods and finite difference methods, was profiled in [8, 11, 13]. Numerical solutions by traditional finite difference and finite element methods are oscillatory and inaccurate in the presence of boundary layers, which is the key difficulty of singularly perturbed problems. Theories for linear singularly perturbed problems in one dimension and in two dimensions when the solutions have only regular layers were discussed in detail by [10]. I The authors were supported in part by NSF of China (No. 11471196, 10971254), and NSF of Shandong Province (No. ZR2016JL004). ∗ Corresponding author Email addresses: [email protected] (Juan Zhang), [email protected] (Huanzhen Chen)

Manuscript

July 12, 2018

Superconvergence approximations of singularly perturbed two-point boundary value problems of reaction-diffusion type and convection-diffusion type were studied with standard finite element method in [4, 19]. The discontinuous Galerkin finite element method was applied to a nonstationary nonlinear convection-diffusion problem and a priori error estimates uniform with respect to the diffusion coefficient and valid even in the purely convective case [5]. The error estimates with maximum norms were investigated in [9]. The authors investigated the optimal control problems by Legendre-Galerkin spectral methods in [21]. Spectral discretizations for singular perturbation problems and an effective iterative method for the solution of spectral systems were proposed in [7]. By introducing additional collocation points, efficient RKN-type Fourier and trigonomeric collocation methods for second-order differential equations were studied in [16, 17], and stabilized treatment of singular perturbation problems with spectral methods was presented in [3]. Spectral methods with interface point were presented to deal with some singularly perturbed problems in ordinary differential equations [6, 15]. An effective Chebyshev pseudospectral method was investigated to resolve the singularly perturbed boundary value problems and accurate results were obtained for very thin boundary layers with a fairly small number of spectral collocation points in [14]. The author developed an a posteriori error estimate for viscoelastic flow calculations using spectral elements in [12]. And a posteriori error estimates for Galerkin spectral methods of Poission equations were discussed in [20, 22]. The authors carried out a posteriori error estimation of Legendre spectral approximations for the Stokes/Darcy coupled problem in [18]. A posteriori error estimates of Legendre Galerkin spectral method for quadratic optimal control problems governed by parabolic equations were studied in [2]. However there is few work concerning on the a posteriori error estimates of spectral methods for singular perturbation problems. The potential difficulty is how to design a lower bound estimate for the numerical solutions. The a posteriori error estimates play central roles in the adaptive techniques. In this work, basing on weighted orthogonal Jacobi polynomials, we approximate the one-dimension singular perturbation problems with spectral methods. Meanwhile, involving the constant eigenvalue, we use weighted orthogonal basis functions to discretize the model. Note that the orthogonal coefficients can be listed with explicit formulae. As a direct consequence, by the relationships between Jacobi extension and estimates of orthogonal coefficients, a lower bounds a posteriori error estimate with L2 -weighted norm is automatically established. The theoretical analysis is sharp. As a byproduct, a posteriori error estimate with weighted H −1 norms is also obtained. 2. Preliminaries Our model problems are singularly perturbed two-point boundary value problems with homogeneous Dirichlet boundary conditions on an interval [−1, 1]. ( − u00 (x) + λu(x) = f (x), in I = (−1, 1), (2.1) u(1) = 0 = u(−1), where  > 0 is the singular perturbation parameter. Usually,  << 1, and λ is the eigenvalue. Define an inner-product hu, viI = (u0 , v 0 )I + λ(u, v)I . Then an equivalent weak formula of (2.1) reads: finding u ∈

(2.2) H01 (I)

∀v ∈ H01 (I).

hu, viI = (f, v)I , 2

such that (2.3)

1 For some given N , we construct an orthogonal polynomial set QN = {qn }N n≥2 ⊂ H0 (I) satisfies

hqn , qm iI = hn δnm , where h·, ·iI is defined as in (2.2). Note that q2 (x) = L2 (x) − L0 (x), q3 (x) = L3 (x) − L1 (x), and h2 = 12λ + 6, h3 = 20λ + 10. 5 21 For n ≥ 4, the recursive relationship reads qn (x) −

2n − 1 2 2(2n − 1) −1,−1 λγn−2 1,1 (x) = qn−2 (x) = Jn (x − 1)Jn−2 (x), hn−2 n−1 2n − 2

(2.4)

and  hn = 2(2n − 1)  +

 2λ 4λ2 2 − , γn = . (2n + 1)(2n − 3) (2n − 3)2 hn−2 2n + 1

(2.5)

Here Li (x) and Jnα,β (x) represent the Legendre and classical Jacobi polynomials, respectively. Obviously, one gets that X X (2.6) u(x) = uˆn qn (x), f (x) = fˆn Jn1,1 (x), n≥2

n≥0

where uˆn =

hu, qn iI ˆ (n + 2)(2n + 3) , fn = (f, Jn1,1 )I,ω1,1 . hn 8(n + 1)

Note that for n ≥ 0, (f, qn+2 )I = hu, qn+2 iI = hn+2 uˆn+2 ,

(2.7)

and (f, qn+2 )I −

λγn 2n + 3 2 4 ˆ (f, qn )I = (f, (x − 1)Jn1,1 ) = − fn , hn 2n + 2 n+2

(2.8)

then we have hn+2 uˆn+2 − γn uˆn = hu, qn+2 iI = −

4 ˆ fn . n+2

(2.9)

Theorem 2.1. For n ≥ 2, there holds hn ≤ 2(2n − 1) +

4(2n − 1)λ . (2n + 1)(2n − 3)

(2.10)

Proof. By (2.5), deleting the negative item, it is a direct conclusion that the upper bound estimate holds.

3

3. Lower bound of a posteriori error estimates The corresponding spectral method is to find uN ∈ QN ⊂ H01 (I) such that huN , vN iI = (f, vN )I ,

∀vN ∈ QN .

(3.1)

To investigate the a posteriori error estimates, we define a quasi-orthogonal projection ΠN u(x) ∈ QN satisfying 0 ((u − ΠN u)0 , vN )I + λ(u − ΠN u, vN )I = 0,

vN ∈ QN ,

and we directly know that 0 0 ((ΠN u)0 , vN )I + λ(ΠN u, vN )I = (u0 , vN )I + λ(u, vN )I = (f, vN )I .

Meanwhile for the numerical solution uN , we have 0 ((uN − ΠN u)0 , vN )I + λ(uN − ΠN u, vN )I = 0.

By selecting vN (x) = uN (x) − ΠN u(x), we declare that uN (x) = ΠN u(x). Meanwhile by qn (x) ∈ QN , we have ΠN u(x) =

N X

uˆn qn (x).

(3.2)

n=2

3.1. The a posteriori error estimator with L2ω−1,−1 -norm Setting X u(x) = v¯n (Ln+2 (x) − Ln (x)), n≥0

one readily gets that

u(x) =

X n≥2

(¯ vn−2 −

λγn v¯n )qn (x). hn

vn }n≥0 , Similarly, we expand the second order derivative of u by Jn1,1 with modified coefficients {ˆ u00 (x) =

X n≥0

vˆn

(2n + 3)(n + 2) 1,1 Jn (x). 2

(3.3)

Now we determine the formula of vˆn . Since (qn+2 −

λγn 00 2(2n + 3) −1,−1 00 (2n + 3)(n + 2) 1,1 (Jn+2 ) = Jn . qn ) = hn n+1 2

(3.4)

Then (3.3) turns to u00 (x) =

X n≥2

vˆn (qn+2 (x) −

X λγn λγn (ˆ vn−2 − qn (x))00 + vˆ0 q200 (x) + vˆ1 q300 (x) = vˆn )qn00 (x). hn hn n≥2

And hence vˆn = v¯n . Then we calculate that kuk2ω−1,−1 =

X n≥0

2(2n + 3) vˆ2 . (n + 1)(n + 2) n 4

(3.5)

In view of (2.6), one gets that vˆn−2 −

λγn vˆn = uˆn , hn

n ≥ 2.

Combination (2.4) and (2.6) yields X 2(2n + 3)2 (n + 2)2 Γ(n + 2)Γ(n + 2) ku00 k2ω1,1 = vˆn2 2n + 3 Γ(n + 1)Γ(n + 3) n≥0 X = 2(n + 1)(n + 2)(2n + 3)ˆ vn2 .

(3.6)

(3.7)

n≥0

We define a truncation operation Π1,1 N such that X 1,1 ΠN f (x) = fˆn Jn1,1 (x). n≥N +1

ˆ , by the relationship between uN and Theorem 3.1. For the solutions u and uN , if N > N ΠN u, we directly get that 2 2 00 2 2 2 kf − Π1,1 N +2 f kω 1,1 .  k(u − ΠN u) kω 1,1 + λ ku − ΠN ukω −1,−1 ,
1 ˆ = λ 2 − 5. where N  2

(3.8)

Proof. To this end, we are at the point to discuss weighted norms of the truncation of f . By (2.9) and (3.6), there hold X X 8(n + 1) 8(n + 1) (n + 2)2 2 kf − Π1,1 fˆn2 = [hn+2 uˆn+2 − λγn uˆn ]2 N +2 f kω 1,1 = (2n + 3)(n + 2) (2n + 3)(n + 2) 16 n≥N +3 n≥N +3   X (n + 1)(n + 2)  λ4 γ 4 2 2 2 ≤4 h2n+2 + 2 n vˆn2 + λ2 γn2 vˆn−2 + λ2 γn+2 vˆn+2 2(2n + 3) hn n≥N +3 X n 2(n + 1)(n + 2) 4(2n + 3)λ 16λ4 ≤ [(2(2n + 3) + )2 + ] 2λ 2 4 2n + 3 (2n + 5)(2n + 1) (2n + 1) [2(2n − 1) + 2n+1 ] n≥N +1 8(2n + 3)(n + 4)λ2 8n(n − 1)λ2 o 2 + vˆ + 2 (2n + 7)(2n + 5) (2n − 1)(2n + 1)2 n    X  8(n + 1)(n + 2) 8(2n + 3)(n + 4) 8n(n − 1) ≤ 32(n + 1)(n + 2)(2n + 3)2 + + + λ2 vˆn2 , 2 2 (2n + 7) 2 (2n − 1) (2n + 3)(2n + 1) (2n + 5) (2n + 1) n≥N +1 where we set

2λ (2n+5)(2n+1)

≤ , i.e., n≥

  12 λ 5 − .  2

Combining (3.5) with (3.7), by 8(n + 1)(n + 2) 8(n + 3)(n + 4) 8n(n − 1) 1 + + ∼ , (2n + 3)(2n + 1)2 (2n + 5)2 (2n + 7) (2n + 1)2 (2n − 1) n

we directly get that (3.8) holds.

This means the weighted truncation error estimate of f can be used as the lower bound of the second-order estimate of u with an additional item, which depends on  and λ. q

Take in mind that, the constraint n & λ means for some smaller , we have to choose the corresponding bigger n to establish this estimate. 5

3.2. The a posteriori error estimator with Hω−1 1,1 -norm Similarly, to investigate the first-order estimate of u − uN , by (3.4), u can be expanded by Jn0,0 with some modified coefficients {˜ vn }n≥0 , X X λγn−2 u(x) = v˜n (Ln (x) − Ln−2 (x)) = v˜n (qn (x) − qn−2 (x)), (3.9) hn−2 n≥2 n≥2 then we declare that

λγn v˜n+2 = uˆn . (3.10) hn ˜ , there holds we directly get that Theorem 3.2. For the solutions u and uN , if N > N X 2 hn uˆ2n = 2 k(u − ΠN u)0 k2 + λku − ΠN uk2 , (3.11) kf − Π1,1 N −2 f kH −1 . v˜n −

ω 1,1

˜= where N

2λ 

+


1 9 2 4

n≥N +1

.

Proof. By (2.6) and (2.10), one knows that 2 kf − Π1,1 N −2 f kH −1 = ω 1,1

X

X 8fˆn2 1 = (hn+2 uˆn+2 − λγn uˆn )2 2 (2n + 3) (n + 2) 2(2n + 3) n≥N −1 n≥N −1 X

1 4λ2 (h2n+2 uˆ2n+2 + uˆ2 ) 2n + 3 (2n + 1)2 n n≥N −1 X 1 λ 4λ [ + ]}hn uˆ2n ≤ {2 + 2n + 1 2n − 3 h n (2n + 1)(2n + 3) n≥N +1 X 16nλ ≤ ]h uˆ2 . [2 + 2 − 9) n n (2n + 1)(4n n≥N +1 ≤

(3.12)

Here we select n such that

i.e.,

16nλ ≤ , (2n + 1)(4n2 − 9) n≥



2λ 9 +  4

 21

,

(3.13)

then by the relationship between uN and ΠN u, (3.12) readily goes to the inequality within (3.11). By direct calculations with (2.6), the identity in (3.11) holds. 4. Conclusion and remarks We have established the lower a posteriori error estimates for one dimensional singularly perturbed eigenvalue problems with spectral methods. Weighted orthogonal Jacobi polynomials are employed to construct the basis functions for spectral approximations. The key within the analysis is an estimation of hn . Our results basically state that approximated truncations of the source item can be selected as error indicators with two norms. Furthermore, with ˆ and N ˜ depending on the  and λ, the a posteriori error estimates are proved as the given N ˆ and N ˜ include boundary layers by the delower bound estimates in details. In fact, the N nominator . By a practical point of view, the lower bound estimate helps us to set a suitable minimum calculation size. For some square-type domains, generalization of the idea to higher dimensions is feasible by tensor products of these basis functions. 6

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