Error estimates of spectral element methods with generalized Jacobi polynomials on an interval

Accepted Manuscript Error estimates of spectral element methods with generalized Jacobi polynomials on an interval

Jianwei Zhou, Juan Zhang, Huantian Xie, Yin Yang

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S0893-9659(17)30080-0 http://dx.doi.org/10.1016/j.aml.2017.03.010 AML 5208

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Applied Mathematics Letters

Received date : 10 January 2017 Revised date : 11 March 2017 Accepted date : 11 March 2017 Please cite this article as: J. Zhou, et al., Error estimates of spectral element methods with generalized Jacobi polynomials on an interval, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.03.010 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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Error estimates of spectral element methods with generalized Jacobi polynomials on an intervalI Jianwei Zhoua , Juan Zhangb,c,∗, Huantian Xiea , Yin Yangd a School of Mathematics and Statistics, Linyi University, Linyi, P.R. China School of Mathematical Sciences, Shandong Normal University, Jinan, P.R. China c Bureau of Statistics of Lanshan District in Linyi City, Linyi, P.R. China d School of Mathematics and Computational Science, Xiangtan University, Xiangtan, P.R. China b

Abstract Based on the generalized Jacobi polynomials with indices (−1, −1), a new upper bound for a posteriori error estimates is proposed, investigated and implemented for generalized Jacobi-Galerkin spectral element approximations. To simplify discussion for the error estimates, the second-order partial differential equation with homogeneous Dirichlet boundary conditions is considered on a unit interval. Keywords: Generalized Jacobi polynomial, weighted orthogonality, spectral element method, a posteriori error estimate 1. Introduction Over the past decades, spectral element methods have been playing increasingly a significant role in engineering computations and scientific simulations. The most important advantage of the spectral element method is that, by some orthogonal properties of given polynomials, the stiff or mass matrix is exactly diagonal. Hence one can use explicit integration schemes to drastically simplify the implementation and reduce the computational cost. The key point is that this method avoids inverting the discretized approximation system. Since Patera [1] proposed the spectral element method, combining the flexibility of the finite element method with the accuracy of spectral techniques, to solve the incompressible Navier-Stokes equations, there have been active research on spectral element methods (see, for instance, [2, 3, 4, 5, 6, 7, 8, 9] and the references cited therein). Zayernouri and Karniadakis [10] employed spectral element methods to get exponentially accurate numerical solutions for fractional ordinary differential equations. In the literatures, most of the researches and studies focused on the classical Jacobi polynomials, denoted by Jkα1 ,α2 (k ≥ 0, α1 , α2 > −1). Shen [11] introduced an efficient dual-PetrovGalerkin method for third and higher odd-order differential equations by Jacobi polynomials with weighted indices (0, 0), i.e. the Legendre polynomials. The authors based on the Jacobi polynomials to study the spectral methods for Volterra integral equations, such as [12, 13] and their references. It is very useful to study numerical methods based on Jacobi polynomials with general negative indices, i.e., the generalized Jacobi polynomials, which are with indices α1 , α2 ∈ R. In fact, the generalized Jacobi polynomials were introduced in [14, 15, 16] with details. The generalized Jacobi polynomials with negative indices not only can be directly used to simplify the numerical analysis for the spectral and spectral element approximations of differential equations, but also lead to very efficient numerical algorithm (see, for instance, [14, 15] and the references therein). I The present research was supported in part by NSFC (No.11571157, 11402108, 11301252, 11671342), PRFEYMS of Shandong Province (No.BS2015DX012, ZR2016AB15) and Guangdong Provincial Engineering Technology Research Center for Data Science (No.2016KF12). ∗ Corresponding author Email addresses: [email protected] (Jianwei Zhou), [email protected] (Juan Zhang), [email protected] (Huantian Xie), [email protected] (Yin Yang)

Preprint submitted to Applied Mathematics Letters

March 11, 2017

Generally, in order to get a numerical solution with acceptable accuracy, one can enhance the numbers of basis and refine the mesh if the a posteriori error indicators are bigger than some given criteria. There are lots of works on hp-version finite element methods focusing on the a posteriori error estimates [17, 18, 19, 20, 21]. However, relatively few studies are devoted to the a posteriori error estimates for spectral element methods. One of the key different points for spectral element methods and hp-version finite element methods is the orthogonal polynomials [22]. The authors [23] presented the a posteriori error indicators with some weighted functions. To minimize the computational cost, one has to depict the optimal a posteriori error indicators. It is with this motivation that we investigate in this paper a posteriori error estimates of spectral element method by generalized Jacobi polynomials with indices α1 , α2 ∈ R. Here we focus on one case of the generalized Jacobi polynomials with (α1 , α2 ) = (−1, −1). The paper is organized as follows. In section 2, we list some preliminaries, which will be used in the following sections. Especially, we introduce generalized Jacobi polynomials with weight indices (−1, −1). In section 3, we consider a second-order equation with homogeneous Dirichlet boundary conditions. Furthermore, we study the relationship between the numerical solution and truncation projection. With the partitions of the interval, we construct discretized schemes of the continuous model problems. In section 4, by the weighted orthogonal operator based on generalized Jacobi polynomials with indices (−1, −1), we derive the a posteriori error estimates with the upper bound for the classical and discretized solutions. Finally, the conclusions and future work are briefly listed in Section 5. 2. Preliminaries We now introduce some notations. Let ω α1 ,α2 (x) = (1 − x)α1 (1 + x)α2 be a positive weight function on I = (−1, 1). One usually requires ω α1 ,α2 (x) ∈ L1 (I), which means α1 , α2 > −1. We use c and C to denote some generic positive constants independent of any polynomial. We denote by L2ωα1 ,α2 (I) the weighted L2 space with inner product Z (v, w)ωα1 ,α2 = v(x)w(x)ω α1 ,α2 (x)dx, I

1 2

and the associated norm kvkωα1 ,α2 = (v, v)ωα1 ,α2 . The index will be omitted when ω α1 ,α2 = 1. As usual, k · k1,I and k · k0,I denote the norm in H 1 (I) and L2 (I), respectively. A . B means there exists a generic positive constant c such that A . cB. Let ( − x, if x ∈ Z− , X (x) = (2.1) 0, others. For k ≥ k0 , X (α1 ) + X (α2 ), we recall the generalized Jacobi polynomials (still denoted by Jkα1 ,α2 ) [16] as     Γ(k + α1 + 1)Γ(k + α2 + 1) x − 1 −α1 −α1 ,α2   Jk+α1 (x), if α1 ∈ Z− , k ≥ X (α1 ),   2  Γ(k + 1)Γ(k + α1 + α2 + 1)      Γ(k + α1 + 1)Γ(k + α2 + 1) x + 1 −α2 α1 ,−α2 Jkα1 ,α2 (x) = Jk+α2 (x), if α2 ∈ Z− , k ≥ X (α2 ),  Γ(k + 1)Γ(k + α1 + α2 + 1) 2    −α1      x + 1 −α2 −α1 ,−α2  x−1  Jk+α1 +α2 (x), if α1 , α2 ∈ Z− , k ≥ X (α1 ) + X (α2 ).  2 2 (2.2) It can be directly got that there holds the orthogonal property (Jsα1 ,α2 (x), Jtα1 ,α2 (x))ωα1 ,α2 = γsα1 ,α2 δs,t ,

2

∀s, t ≥ k0

(2.3)

where δs,t is the Kronecker delta function, and γsα1 ,α2 , kJsα1 ,α2 (x)kωα1 ,α2 =

2α1 +α2 +1 Γ(s + α1 + 1)Γ(s + α2 + 1) . (2s + α1 + α2 + 1)Γ(s + 1)Γ(s + α1 + α2 + 1)

(2.4)

The recurrence relation of the generalized Jacobi polynomial reads k + α1 + α2 + 1 α1 +1,α2 +1 d α1 ,α2 (x) = J Jk−1 (x), dx k 2

−k − α1 − α2 ∈ / {1, 2, . . . , k}.

(2.5)

Here the constraint −k − α1 − α2 ∈ / {1, 2, · · · , k} guarantees that the polynomial Jkα1 ,α2 (x) is k-order without degenerations [22]. Furthermore

2 1 . , Jkα1 ,α2 (−1) = (−1)k k+α Jkα1 ,α2 (x) = (−1)k Jkα2 ,α1 (−x), Jkα1 ,α2 (1) = k+α k k

In this paper, we focus on Jk−1,−1 (x), i.e., (α1 , α2 ) = (−1, −1). For ∀x ∈ I, we list the generalized Jacobi polynomials as J0−1,−1 (x) = 1, J1−1,−1 (x) = x, Jk−1,−1 (x) =

x − 1 x + 1 1,1 Jk−2 (x), k ≥ 2. 2 2

(2.6)

By (2.3) and (2.4), it is easy to calculate that (Jk−1,−1 (x), Jt−1,−1 (x))ω−1,−1 = We define an inner product of H 1 (I) as hv, wiω−1,−1 = (

k−1 δk,t . 2k(2k − 1)

d d v, w) −1,−1 + v(1)w(1) + v(−1)w(−1) dx dx ω

∀v, w ∈ H 1 (I).

(2.7)

By simple calculations and the extended J0−1,−1 (x), J1−1,−1 (x), it is a direct conclusion that {Jk−1,−1 (x)}+∞ k=0 is one of the orthogonal basis functions of H 1 (I), and there holds   (k − 1)2 (1 − δk,0 − δk,1 ) δk,t . hJk−1,−1 (x), Jt−1,−1 (x)iω−1,−1 = 2δk,0 + 4δk,1 + 4k − 2

And hence, for ∀v ∈ H 1 (I), we have v(x) =

∞ X

v˜k Jk−1,−1 (x),

v˜k =

k=0

hv(x), Jk−1,−1 (x)iω−1,−1 . −1,−1 hJk (x), Jk−1,−1 (x)iω−1,−1

Denote by PN the collection of all the generalized Jacobi polynomials with (α1 , α2 ) = (−1, −1) −1,−1 : L2ω−1,−1 (I) 7→ PN (I) ∩ and total degree ≤ N . We now define an orthogonal projection ΠN 2 Lω−1,−1 (I), N X (2.8) v˜k Jk−1,−1 (x). v(x) = Π−1,−1 N k=0

Hereafter, thanks to {Jk−1,−1 (x)}, one can state that

PN (I) = {Jk−1,−1 (x), x ∈ I}N k=0 .

Following the orthogonal property of Jk−1,−1 (x), we have Π−1,−1 N

−1,−1 v, wiω−1,−1 = 0, hv − ΠN

∀w ∈ PN (I),

is a H 1 -orthogonal projection. which means Note that, an important property of {Jk−1,−1 (x)}+∞ k=0 is that Jk−1,−1 (1) = 0,

Jk−1,−1 (−1) = 0.

Hence, {Jk−1,−1 (x)}+∞ k=0 are natural candidates as basis functions for partial differential equations with homogeneous Dirichlet boundary conditions. Meanwhile, we directly state that PN (I)∩L2ω−1,−1 (I) = PN (I) ∩ H01 (I). 3

3. The model and its spectral element approximation Without loss of generality, we consider the Poisson equation with homogeneous Dirichlet boundary condition  d2  − 2 u(x) = f (x), x ∈ I, (3.1) dx  u(±1) = 0. As we all known, this equation has a unique solution. An equivalent weak formula of (3.1) reads: Finding u ∈ H01 (I) such that (

d d u, v) = (f, v), dx dx

∀v ∈ H01 (I).

(3.2)

Here we introduce some notations which can be extended into multi-intervals with an affine mapping. −1,−1 Definition 3.1. For ∀v ∈ H 1 (I), we difine a global orthogonal projection ΠN,I : H 1 (I) 7→ PN (I) as Z x Π0,0 Π−1,−1 v = N −1,I ∂x v(t)dt + v(−1), N,I −1

where Π0,0 N −1,I w(x) =

N −1 X

w ˆi Ji0,0 (x),

w ˆi =

i=0

(w, Ji0,0 ) kJi0,0 k20,I

.

Lemma 3.1. For ∀v ∈ H 1 (I), there holds Π−1,−1 N,I v(±1) = v(±1). Proof. Furthermore, for N ≥ 2, Π−1,−1 N,I v(x) = =

Z

x

−1

=

N X i=1

Z

vˆi

x

−1

Π0,0 N −1,I ∂x v(t)dt + v(−1) =

Z

x

−1

Π0,0 N −1,I

(3.3)

∂x

∞ X

vˆi Ji−1,−1 (x)

i=0

!!

dt + v(−1)

! N   X i − 1 0,0 Ji−1 (x) dt + v(−1) = vˆi Ji−1,−1 (x) − Ji−1,−1 (−1) + v(−1) 2

(3.4)

i=1

N

X v(1) − v(−1) (x + 1) + vˆi Ji−1,−1 (x) + v(−1). 2 i=2

In the light of (3.4), and by the definition listed in (2.6), it is obvious that Π−1,−1 N,I v(±1) = v(±1). Also, we can reformulate (3.4) as Π−1,−1 N,I v(x) =

N X

vˆi Ji−1,−1 (x).

(3.5)

i=0

Remark 3.1. Combining (2.5) with the inner-product defined in (2.7), we immediately conclude that the result stated in (3.5) is consistent with (2.8). By (3.5), it is a direct conclusion that the projection Π−1,−1 is the truncation according {Jk−1,−1 (x)}∞ k=0 . N,I Lemma 3.2. For ∀v ∈ H 1 (I), there holds 0 0 ((v − Π−1,−1 N,I v) , w ) = 0,

4

∀w ∈ PN (I).

(3.6)

Proof. For ∀v ∈ H 1 (I), one immediately derives that 0,0 0 0 0 0 0 ((v − Π−1,−1 N,I v) , w ) = (v − ΠN −1,I v , w ) = 0,

∀w ∈ PN (I),

since w0 ∈ PN −1 (I), which is the desired result (3.6). Lemma 3.3. If vN is a numerical solution of (3.2), we declare that vN = Π−1,−1 N,I v,

(3.7)

Proof. For the global case (i.e., the spectral method), with the orthogonal property of the projection, we get that (

d d d d (Π−1,−1 v), wN ) = ( v, wN ) = (f, wN ), dx N,I dx dx dx

∀wN ∈ PN (I),

And then we conclude that the relationship (3.7) between the numerical solution and the classical solution holds, where we used the uniqueness of the numerical approximation for (3.2). Now we are at the point to introduce the discretized approximations of (3.1). Define the partition T , {−1 = x0 < x1 < · · · < xn = 1} and sub-interval (called element) Ii = (xi−1 , xi ), hi = |xi − xi−1 |, for 1 ≤ i ≤ n. Let δ = δ(h, M ), where h = max hi and M = min Ni . There obviously Ii ∈T

Ii ∈T

holds δ → 0 when h → 0 and/or M → +∞. Generally, it can be stated by δ = O(h/M ). Let Pδ be the collection of all algebraic polynomial with indices (Ni , Ii ), i.e., PNi on Ii . The approximation space is Vδ = {v ∈ H01 (I) : v|Ii ◦ Fi ∈ PNi (I), ∀Ii ∈ T }, where Fi : Ii 7→ I is an affine mapping. And herein, the approximation scheme for (3.1) reads: finding uδ ∈ Vδ such that (

d d uδ , vδ ) = (f, vδ ), dx dx

∀vδ ∈ Vδ .

(3.8)

−1,−1 0,0 and Π0,0 The projector Π−1,−1 N,I with the affine mapping Fi Ni ,Ii and ΠNi ,Ii are deduced from Πδ on each element Ii , respectively.

Lemma 3.4. ([17]) For any v ∈ L2 (I) and v|Ii ∈ H si (Ii ), there holds kv − Π−1,−1 Ni ,Ii vkq,Ii .

hµi i −q

Nisi −q

kvksi ,Ii ,

∀Ii ∈ T ,

(3.9)

where µi = min{Ni + 1, si } and 0 ≤ q ≤ si . 4. The a posteriori error estimates From now on, we employ weighted orthogonal properties of generalized Jacobi polynomials to investigate the a posteriori error estimates for (3.1). We list our main result in the following theorem, which states the upper bound of the a posteriori error estimates. Theorem 4.1. Let u and uδ be the solutions of (3.1) and (3.8), respectively. Then there holds ku − uδ k1,I .

"

n  X hi i=1

Ni

0,0 kΠN f+ i ,Ii

d2 uδ kIi dx2

2 # 21

5

+

" n  X hi i=1

Ni

0,0 kf − ΠN f kIi i ,Ii

2 # 12

.

(4.1)

Proof. For ∀v ∈ H01 (I), in the light of (3.8) and (3.9), d d d d d2 (u − uδ ), v) = ( (u − uδ ), (v − Πδ−1,−1 v)) = (− 2 (u − uδ ), v − Πδ−1,−1 v) dx dx dx dx dx d2 0,0 −1,−1 0,0 −1,−1 = (Πδ f + 2 uδ , v − Πδ v) + (f − Πδ f, v − Πδ v) dx n 2 X d 0,0 −1,−1 ≤ uδ k0,Ii kv − Π−1,−1 {kΠ0,0 Ni ,Ii vk0,Ii + kf − ΠNi ,Ii f k0,Ii kv − ΠNi ,Ii vk0,Ii } Ni ,Ii f + dx2

(

i=1

(4.2)

n X d2 hi hi ≤ {kΠ0,0 uδ k0,Ii kvk1,Ii + kf − Π0,0 kvk1,Ii } Ni ,Ii f + Ni ,Ii f k0,Ii dx2 Ni Ni i=1 "  2 # 12  2 # 12 "X n  n   X hi d2 hi 0,0 0,0 ≤ kΠ f + 2 uδ k0,Ii + kf − ΠNi ,Ii f k0,Ii kvk1,I .   Ni Ni ,Ii dx Ni i=1

i=1

By the definition of H 1 -semi-norm and (4.2), one readily goes to k

d (u − uδ )k0,I = dx

.

"

n  X hi i=1

Ni

sup dv ∈L2 (I) dx

kΠ0,0 Ni ,Ii f

d d ( dx (u − uN ), dx v) dv k dx k0,I 2 # 1

d2 + 2 uδ k0,Ii dx

2

+

" n  X hi i=1

Ni

kf −

Π0,0 Ni ,Ii f k0,Ii

2 # 21

.

Combining the above analyses with Poincar´e inequality, we declare that (4.1) holds. Remark 4.1. To prove the above theorem, we set q = 0, si = 1 in (3.9). 5. Conclusion In this paper, we investigate the a posteriori error estimates of generalized Jacobi-Galerkin spectral element methods for solving second-order partial differential equations. In the light of the second-order derivative, we state in the first remark that the projection is the numerical solution step by step. From elements to the unit interval, there exists the affine mapping Fi such that the global solution consisted of piecewise local numerical solution with combinations. Nevertheless, we omit to study the lower bound estimates, because the spectral element method is with higher accuracy with finite polynomials. The tensor product can be used to extend this technique to study the two-dimensional cases. Both the problem focusing on the truncation errors and adaptive p-version finite element methods with this kind of estimates are included in our ongoing work. References [1] A.T. Patera, A spectral element method for fluid dynamics: Laminar flow in a channel expansion, Journal of Computational Physics, 54(1984)468-488. [2] C. Bernardi, Y. Maday, Spectral methods, in Handbook of Numerical Analysis, P.G. Ciarlet and J.-L. Lions, eds., Elsevier, Amsterdam, (1997)209-486. [3] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral methods in fluid dynamics, Springer-Verlag, (1988). [4] X.M. Xiang, Numerical analysis of spectral methods, Science Press, Beijing, (2000). [5] L.W. Ho and A. T. Patera, A legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows, Computer Methods in Applied Mechanics & Engineering, 80(1990)355-366. 6

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