Mixed Fourier–Jacobi spectral method

J. Math. Anal. Appl. 315 (2006) 8–28 www.elsevier.com/locate/jmaa

Mixed Fourier–Jacobi spectral method Wang Li-lian, Guo Ben-yu ∗,1 Department of Mathematics, Division of Computational Science of E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China Received 20 May 2003 Available online 9 November 2005 Submitted by W.L. Wendland

Abstract This paper is for mixed Fourier–Jacobi approximation and its applications to numerical solutions of semiperiodic singular problems, semi-periodic problems on unbounded domains and axisymmetric domains, and exterior problems. The stability and convergence of proposed spectral schemes are proved. Numerical results demonstrate the efficiency of this new approach.  2005 Elsevier Inc. All rights reserved. Keywords: Mixed Fourier–Jacobi spectral method; Applications

1. Introduction In this paper, we investigate the mixed Fourier–Jacobi spectral method and its applications. This work is motivated by several facts. Firstly we need to solve semi-periodic problems numerically when we study the boundary layer, the flow past a suddenly heated vertical plate and other related problems, see [9,11,12]. We usually use the Fourier spectral-finite difference method, the Fourier spectral-finite element method, and the mixed Fourier–Legendre and Fourier–Chebyshev spectral methods, see [2,5,8–10,12]. However in many cases, we have to consider semi-periodic singular problems. In this case, it is natural to use the Jacobi approximation developed in [3,4,6], * Corresponding author.

E-mail address: [email protected] (B.Y. Guo). 1 The work of this author is supported in part by NSF of China, No. 10471095, SF of Shanghai No. 04JC14062, The

Fund of Chinese Education Ministry No. 20040270002, The Shanghai Leading Academic Discipline Project No. T0401, The Funds for E-institutes of Universities No. E03004, and The Found No. 04DB15 of Shanghai Municipal Education Commission. 0022-247X/$ – see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.09.088

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

9

in the non-periodic directions. The second motivation is numerical simulation of semi-periodic problems on unbounded domains. We may impose certain artificial boundary conditions in the non-periodic directions. However, it causes additional errors. In particular, it is difficult to apply this trick to nonlinear problems. Some authors developed the mixed Fourier–Laguerre and Fourier–Hermite spectral methods. But both of them require quadratures over unbounded domains. In fact, we could reform such problems and then use the mixed Fourier–Jacobi spectral method for the resulting ones. Finally, the mixed Fourier–Jacobi spectral method is also motivated by some problems on axisymmetric domains, see [1], and exterior problems. This paper is organized as follows. In the next section, we recall some basic results on one-dimensional Jacobi approximation. In Section 3, we establish some results on the mixed Fourier–Jacobi approximation which play important roles in the analysis of mixed Fourier–Jacobi spectral method. Section 4 is for some applications. We first consider a linear model problem, and then deal with the nonlinear Klein–Gordon equation. The stability and convergence of proposed schemes are proved. Some other applications are discussed. We also present some numerical results showing the high accuracy of this new approach. 2. Preliminaries Let Λ = (−1, 1) and χ(x) be a certain weight function in the usual sense. For 1
p
∞, p Lχ (Λ) = {v | v is measurable and vLpχ < ∞}, with the norm
 1/p p , 1
p < ∞, Λ |v(x)| χ(x) dx vLpχ = ess supx∈Λ |v(x)|, p = ∞. In particular, we denote by (u, v)χ and vχ the inner product and the norm of space L2χ (Λ). Furthermore, for non-negative integer m, Hχm (Λ) = {v | vm,χ < ∞}, equipped with the semi 2 1/2 . For any real r > 0, we define norm |v|m,χ = ∂xm vχ , and the norm vm,χ = ( m k=0 |v|k,χ ) r r the space Hχ (Λ) by the space interpolation. H0,χ (Λ) is the closure in Hχr (Λ) of the set of all infinitely differentiable functions with compact support in Λ. We omit r for r = 0 and omit χ for χ ≡ 1, in the notations. (α,β) We denote by Jl (x) the Jacobi polynomials of degree l. Let χ (α,β) (x) = (1 − x)α (1 + x)β , α, β > −1. The set of Jacobi polynomials is an L2χ (α,β) (Λ)-orthogonal system. For any positive integer M, PM (Λ) stands for the set of all algebraic polynomials of degree 0 (Λ) = {v | v ∈ P (Λ), at most M. Further, 0 PM (Λ) = {v | v ∈ PM (Λ), v(−1) = 0} and PM M v(−1) = v(1) = 0}. We shall use two inverse inequalities in PM (Λ), stated below. They are proved in [4]. In the sequel, c denotes a generic positive constant independent of any function and M. Lemma 2.1. For any φ ∈ PM (Λ) and 1
p
q
∞, φLq

χ (α,β)

,Λ


cM σ (α,β)(1/p−1/q) φLp

χ (α,β)

,Λ ,

where σ (α, β) = 2 max(α, β) + 2 if max(α, β)  − 12 , and σ (α, β) = 1 otherwise. Lemma 2.2. For any φ ∈ PM (Λ) and r  0, φr,χ (α,β) ,Λ
cM 2r φχ (α,β) ,Λ .

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If, in addition, α, β > r − 1, then φr,χ (α,β) ,Λ
cM r φχ (α−r,β−r) ,Λ . We shall use also several orthogonal projections in the next section. For technical reasons, we define the non-uniformly weighted Sobolev space Hχr (α,β) ,A (Λ). For any integer r  0,   Hχr (α,β) ,A (Λ) = v | v is measurable and vr,χ (α,β) ,A,Λ < ∞ , equipped with the following semi-norm and norm: |v|r,χ (α,β) ,A,Λ

  = ∂ r v  x

χ (α+r,β+r) ,Λ

,

vr,χ (α,β) ,A,Λ =

r

1/2 |v|2k,χ (α,β) ,A,Λ

.

k=0

For any real number r > 0, we define the space Hχr (α,β) ,A (Λ) by space interpolation. Further, let Hχr (α,β) ,∗ (Λ) = {v | ∂x v ∈ Hχr−1 (α,β) ,A (Λ)} with the semi-norm |v|r,χ (α,β) ,∗,Λ = |∂x v|r−1,χ (α,β) ,A,Λ and the norm vr,χ (α,β) ,∗,Λ = ∂x vr−1,χ (α,β) ,A,Λ . Let (u, v)χ (α,β) ,Λ be the inner product of the space L2χ (α,β) (Λ). The orthogonal projection PM,α,β,Λ : L2χ (α,β) (Λ) → PM (Λ) is defined by (PM,α,β,Λ v − v, φ)χ (α,β) ,Λ = 0,

∀φ ∈ PM (Λ).

We have the following basic result, see [4, p. 387]. Lemma 2.3. For any v ∈ Hχr (α,β) ,A (Λ) and integer r  0, PM,α,β,Λ v − vχ (α,β) ,Λ
cM −r |v|r,χ (α,β) ,A,Λ . µ

Next, we take α, β, γ , δ > −1 and introduce the spaces Hα,β,γ ,δ (Λ), 0
µ
1. For µ = 0, 0 2 Hα,β,γ ,δ (Λ) = Lχ (γ ,δ) (Λ). For µ = 1,   1 Hα,β,γ ,δ (Λ) = v | v is measurable and v1,α,β,γ ,δ,Λ < ∞ equipped with the following semi-norm and norm: |v|1,α,β,γ ,δ,Λ = ∂x vχ (α,β) ,Λ ,

1/2  v1,α,β,γ ,δ,Λ = |v|21,α,β,γ ,δ,Λ + v2χ (γ ,δ) ,Λ .

µ

For 0 < µ < 1, the space Hα,β,γ ,δ (Λ) is defined by space interpolation. In particular,   1 1 0 Hα,β,γ ,δ (Λ) = v v ∈ Hα,β,γ ,δ (Λ) and v(−1) = 0 . Let aα,β,γ ,δ (u, v) = (∂x u, ∂x v)χ (α,β) ,Λ + (u, v)χ (γ ,δ) ,Λ . 1 1 The orthogonal projection PM,α,β,γ ,δ,Λ : Hα,β,γ ,δ (Λ) → PM (Λ) is defined by  1  aα,β,γ ,δ PN,α,β,γ ,δ,Λ v − v, φ = 0, ∀φ ∈ PM (Λ). 1 1 The orthogonal projection 0 PM,α,β,γ ,δ,Λ : 0 Hα,β,γ ,δ (Λ) → 0 PM (Λ) is defined by   1 aα,β,γ ,δ 0 PM,α,β,γ ,δ,Λ v − v, φ = 0, ∀φ ∈ 0 PM (Λ).

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

11

1,0 0 1 The orthogonal projection P˜M,α,β,Λ : H0,χ (α,β) (Λ) → PM (Λ) is defined by     1,0 0 (Λ). ∂x P˜M,α,β,Λ v − v , ∂x φ χ (α,β) ,Λ = 0, ∀φ ∈ PM

We shall use the following results, see [7]. Lemma 2.4. If α
γ + 2 and β
δ + 2, then for any v ∈ Hχr (α,β) ,∗ (Λ) and integer r  1,   1 1−r  P |v|r,χ (α,β) ,∗,Λ . M,α,β,γ ,δ,Λ v − v 1,α,β,γ ,δ,Λ
cM If, in addition, α
γ + 1, β
δ + 1, then for 0
µ
1,   1 P v − v
cM µ−r |v| (α,β) M,α,β,γ ,δ,Λ

r,χ

µ,α,β,γ ,δ,Λ

,∗,Λ .

Lemma 2.5. If α
γ + 1, β
δ + 2, 0 < α < 1, β < 1 or α
γ + 2, β
0, δ  0, then for any 1 r v ∈ 0 Hα,β,γ ,δ (Λ) ∩ Hχ (α,β) ,∗ (Λ) and integer r  1,   1 1−r   0P |v|r,χ (α,β) ,∗,Λ . M,α,β,γ ,δ,Λ v − v 1,α,β,γ ,δ,Λ
cM If, in addition, β
δ + 1, then for 0
µ
1,  1   0P v − v
cM µ−r |v| M,α,β,γ ,δ,Λ

r,χ (α,β) ,∗,Λ .

µ,α,β,γ ,δ,Λ

1 r Lemma 2.6. If −1 < α, β < 1, then for any v ∈ H0,χ (α,β) (Λ) ∩ Hχ (α,β) ,∗ (Λ) and r  1,   1,0 1−r  P˜ |v|r,χ (α,β) ,∗,Λ . M,α,β,Λ v − v 1,χ (α,β) ,Λ
cM

If, in addition, −1 < α, β
0 or 0 < α, β < 1, then for all 0
µ
1,   1,0 P˜ v − v  (α,β)
cM µ−r |v| (α,β) . M,α,β,Λ

µ,χ

,Λ

r,χ

,∗,Λ

3. Mixed Fourier–Jacobi approximation In this section, we study the mixed Fourier–Jacobi approximation. Let x = (x1 , x2 ), Λ1 = {x1 | −1 < x1 < 1}, Λ2 = {x2 | 0 < x2 < 2π} and Ω = Λ1 × Λ2 . For r  0, we define the r (Ω) in the usual way. For simplicity, we denote their semiweighted spaces Hχr (Ω) and H0,χ p norm and norm by |v|r,χ and vr,χ , respectively. vLpχ denotes the norm of Lχ (Ω), 1
p
∞. In addition, (u, v)r,χ denotes the inner product of Hχr (Ω) for integer r. We omit r or χ in the notations when r = 0 or χ ≡ 1, respectively. Let M and N be any positive integers. VN (Λ2 ) is the set of all real-valued trigonometric polynomials of degree at most N on Λ2 . VM,N = PM (Λ1 ) ⊗ VN (Λ2 ), 0 VM,N = 0 PM (Λ1 ) ⊗ 0 0 (Λ ) ⊗ V (Λ ). We first establish two inverse inequalities in V = PM VN (Λ2 ) and VM,N 1 N 2 M,N . Theorem 3.1. For any φ ∈ VM,N and 2
p
∞,  1/2−1/p φLp
c M σ (γ ,δ) N φχ (γ ,δ) . χ (γ ,δ)

Proof. Let 1 φl (x1 ) = 2π

Λ2

φ(x1 , x2 )e−ilx2 dx2 .

(3.1)

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L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

Clearly φl ∈ PM (Λ1 ). By the orthogonality of trigonometric functions,

φ2χ (γ ,δ) = 2π φl 2χ (γ ,δ) ,Λ . 1

|l|
N

The above with Lemma 2.1 leads to



 1/2 1 φL∞ (Ω)
φl ∞,Λ1
cM 2 σ (γ ,δ) φl χ (γ ,δ) ,Λ1
c M σ (γ ,δ) N φχ (γ ,δ) . |l|
N

|l|
N

Accordingly p

φLp

χ (γ ,δ)

 p/2−1 p−2 p
φL∞ φ2χ (γ ,δ)
c M σ (γ ,δ) N φχ (γ ,δ) .

2

We now establish another inverse inequality. Let αi , βi > −1, i = 1, 2, and χ (α1 ,β1 ) (x) = (α2 ,β2 ) (x) = χ (α2 ,β2 ) (x ). Further, let α = (α , α ), β = (β , β ), and 1 ), χ 1 1 2 1 2     r,s 2 r s 2 Hχ (α,β) (Ω) = L Λ2 ; Hχ (α1 ,β1 ) (Λ1 ) ∩ H Λ2 ; Lχ (α2 ,β2 ) (Λ1 )

χ (α1 ,β1 ) (x

with the norm vH r,s

χ (α,β)

 = v2L2 (Λ

r 2 ;H (α ,β ) (Λ1 )) χ 1 1

+ v2H s (Λ

1/2 2 2 ;L (α ,β ) (Λ1 )) χ 2 2

.

Theorem 3.2. For any φ ∈ VM,N and r, s  0,   φH r,s
c M 2r + N s φχ (η1 ,η2 ) , χ (α,β)

where η1 = min(α1 , α2 ) and η2 = min(β1 , β2 ). If, in addition, α1 , β1 > r − 1, then   φH r,s
c M r + N s φχ (θ1 ,θ2 ) , χ (α,β)

where θ1 = min(α1 − r, α2 ) and θ2 = min(β1 − r, β2 ). Proof. Let φl (x1 ) be the same as in (3.1). By Lemma 2.2, for any integers 0
µ
r,

  µ 2  ∂ φ  (α ,β ) = 2π ∂ µ φl 2 (α ,β )
cM 4µ φl 2χ (α1 ,β1 ) ,Λ x1 x1 χ 1 1 χ 1 1 ,Λ 1

|l|
N

|l|
N

1


cM 4µ φ2χ (η1 ,η2 ) . Similarly, for any integers 0
ν
s, ∂xν2 φ2 (α2 ,β2 )
cN 2ν φ2 (η1 ,η2 ) . The previous statements χ χ with space interpolation lead to the first result. We next prove the second result. Due to α1 , β1 > r − 1, we use Lemma 2.2 to obtain that for integer µ  0,

 µ 2 ∂ φ  (α ,β )
cM 2µ φl 2χ (α1 −r,β1 −r) ,Λ
cM 2µ φ2χ (θ1 ,θ2 ) . x1 1 1 χ 1

|l|
N

The rest part of the proof is clear.

2

We now turn to some orthogonal projections. To do this, we introduce the space     Hχr,s(γ ,δ) ,A (Ω) = L2 Λ2 ; Hχr (γ ,δ) ,A (Λ1 ) ∩ H s Λ2 ; L2χ (γ ,δ) (Λ1 ) , r, s  0,

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

with the norm vH r,s

χ (γ ,δ) ,A

 = v2L2 (Λ

r 2 ;H (γ ,δ) (Λ1 )) χ

,A

+ v2H s (Λ

1/2 2 2 ;L (γ ,δ) (Λ1 )) χ

13

.

r,s r,s Denote by Hp,χ (γ ,δ) ,A (Ω) the subspace of Hχ (γ ,δ) ,A (Ω), consisting of functions whose derivatives of order up to r − 1 has the period 2π for the variable x2 . Throughout this paper, all spaces with the subscript p have the similar meanings. The orthogonal projection PM,N,γ ,δ : L2p,χ (γ ,δ) (Ω) → VM,N is defined by

(PM,N,γ ,δ v − v, φ)χ (γ ,δ) = 0,

∀φ ∈ VM,N .

r,s Theorem 3.3. For any v ∈ Hp,χ (γ ,δ) ,A (Ω) and integers r, s  0,   PM,N,γ ,δ v − vχ (γ ,δ)
c M −r + N −s vH r,s . χ (γ ,δ) ,A

Proof. Let vl (x1 ) =

1 2π

v(x1 , x2 )e−ilx2 dx2 ,

(3.2)

Λ2

and PM,γ ,δ,Λ1 be the L2χ (γ ,δ) (Λ1 )-orthogonal projection as defined in Section 2. Clearly,

  PM,N,γ ,δ v(x) = PM,γ ,δ,Λ1 vl (x1 ) eilx2 . |l|
N

By Lemma 2.3 and the result on the Fourier approximation, PM,N,γ ,δ v − v2χ (γ ,δ)



= 2π PM,γ ,δ,Λ1 vl − vl 2χ (γ ,δ) ,Λ + 2π vl 2χ (γ ,δ) ,Λ 1

|l|
N


cM −2r v2L2 (Λ

r 2 ;H (γ ,δ) (Λ1 )) χ ,A

|l|>N

+ cN −2s v2H s (Λ

1

2 2 ;L (γ ,δ) (Λ1 )) χ

.

2

In many cases, the coefficients of differential equations degenerate in different ways. Thus we need to consider the related orthogonal projections in certain non-uniformly weighted Sobolev 0 2 spaces. For this purpose, let Hα,β,γ ,δ (Ω) = Lχ (γ ,δ) (Ω), and define   1 Hα,β,γ ,δ (Ω) = v | v is measurable on Ω and v1,α,β,γ ,δ < ∞ , equipped with the following semi-norm and norm: 1/2  , |v|1,α,β,γ ,δ = ∂x1 v2χ (α1 ,β1 ) + ∂x2 v2χ (α2 ,β2 )   2 1/2 . v1,α,β,γ ,δ = |v|1,α,β,γ ,δ + v2χ (γ ,δ) µ

For 0 < µ < 1, the space Hα,β,γ ,δ (Ω) is defined by space interpolation. In particular, 1 1 1 0 Hα,β,γ ,δ (Ω) = {v | v ∈ Hα,β,γ ,δ (Ω) and v(−1, x2 ) = 0 for x2 ∈ Λ2 }, and H0,α,β,γ ,δ (Ω) = {v | 1 1 v ∈ Hα,β,γ ,δ (Ω) and v(−1, x2 ) = v(1, x2 ) = 0 for x2 ∈ Λ2 }. The space Hp,α,β,γ ,δ (Ω) is defined as before.

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L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

Now, let aα,β,γ ,δ (u, v) = (∂x1 u, ∂x1 v)χ (α1 ,β1 ) + (∂x2 u, ∂x2 v)χ (α2 ,β2 ) + (u, v)χ (γ ,δ) . 1 1 The orthogonal projection PM,N,α,β,γ ,δ : Hp,α,β,γ ,δ (Ω) → VM,N is defined by  1  aα,β,γ ,δ PM,N,α,β,γ ,δ v − v, φ = 0, ∀φ ∈ VM,N . 1 1 The orthogonal projection 0 PM,N,α,β,γ ,δ : 0 Hp,α,β,γ ,δ (Ω) → 0 VM,N is defined by  1  aα,β,γ ,δ 0 PM,N,α,β,γ ,δ v − v, φ = 0, ∀φ ∈ 0 VM,N . 1,0 0 1 1 The orthogonal projection P˜M,N,α,β,γ ,δ : H0,α,β,γ ,δ (Ω) ∩ Hp,α,β,γ ,δ (Ω) → VM,N is defined by

  1,0     1,0   ∂x1 P˜M,N,α,β,γ ,δ v − v , ∂x1 φ χ (α1 ,β1 ) + ∂x2 P˜M,N,α,β,γ ,δ v − v , ∂x2 φ χ (α2 ,β2 ) = 0, 0 . ∀φ ∈ VM,N

For description of approximation results, we define the non-isotropic space     r,s,σ r+σ −2 2 r 1 Mα,β,γ ,δ (Ω) = L Λ2 ; Hχ (α1 ,β1 ) ,∗ (Λ1 ) ∩ H Λ2 ; Hχ (α1 ,β1 ) ,∗ (Λ1 )     ∩ H s−1 Λ2 ; Hα11 ,β1 ,γ ,δ (Λ1 ) ∩ H s Λ2 ; L2χ (α2 ,β2 ) (Λ1 ) , where r, s  1, σ = 1 or 2. Its norm vM r,s,σ r,s,σ Mp,α,β,γ ,δ (Ω)

α,β,γ ,δ

is defined in the usual way. The meaning of

is clear. One of the main results in this section is stated below.

Theorem 3.4. If α1
γ + σ,

β1
δ + σ,

γ
α2 , δ
β2 , σ = 1 or 2,

(3.3)

then for any v and integers r, s  1,   1   1−r  P + N 1−s vM r,s,σ . M,N,α,β,γ ,δ v − v 1,α,β,γ ,δ
c M

(3.4)

r,s,σ ∈ Mp,α,β,γ ,δ (Ω)

α,β,γ ,δ

If, in addition, σ = 1, then for all 0
µ
1,   1  P M,N,α,β,γ ,δ v − v µ,α,β,γ ,δ µ  −r   1−µ M + N −s 1 + M −1 N
c M 1−r + N 1−s vM r,s,1 .

(3.5)

α,β,γ ,δ

1 Proof. Let vl (x1 ) be the same as in (3.2), and PM,α be the Hα11 ,β1 ,γ ,δ (Λ1 )-orthogonal 1 ,β1 ,γ ,δ,Λ1 projection as defined in Section 2. Set

  1 PM,α v (x1 ) eilx2 ∈ VM,N . v ∗ (x) = 1 ,β1 ,γ ,δ,Λ1 l |l|
N

By the projection theorem, (3.3), Lemma 2.4 and the result on the Fourier approximation, we obtain that for σ = 2, r  1 (or σ = 1, r  2) and s  1, 2  1  P M,N,α,β,γ ,δ v − v 1,α,β,γ ,δ =

inf φ − v21,α,β,γ ,δ
v ∗ − v21,α,β,γ ,δ

φ∈VM,N

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28


c

 P 1 |l|
N

M,α1 ,β1 ,γ ,δ,Λ1 vl

2 − vl 1,α

1 ,β1 ,γ ,δ,Λ1

15

 1 2  + l 2 PM,α v − vl χ (γ ,δ) ,Λ 1 ,β1 ,γ ,δ,Λ1 l 1

  +c vl 21,α1 ,β1 ,γ ,δ,Λ1 + l 2 vl 2χ (α2 ,β2 ) ,Λ 1

|l|>N


cM 2−2r v2L2 (Λ + cN

2−2s

r 2 ;H (α ,β ) (Λ1 )) χ 1 1 ,∗

 v2H s−1 (Λ

1 2 ;Hα ,β 1 1



+ v2

−2 H 1 (Λ2 ;H r+σ (α ,β ) (Λ1 )) χ

1 1 ,∗

+ v2H s (Λ ;L2 2 ,γ ,δ (Λ1 )) (α χ

 2
c M 1−r + N 1−s v2M r,s,σ .

2 ,β2



) (Λ1 ))

α,β,γ ,δ

If r = 1, σ = 1 and s  1, then we obtain from the projection theorem that 2  1 2 2  P M,N,α,β,γ ,δ v − v 1,α,β,γ ,δ
v1,α,β,γ ,δ
cvM r,s,σ . α,β,γ ,δ

We now use an duality argument to prove (3.5) with σ = 1 and µ = 0. Let al,α,β,γ ,δ (u, v) = (∂x1 u, ∂x1 v)χ (α1 ,β1 ) ,Λ1 + l 2 (u, v)χ (α2 ,β2 ) ,Λ1 + (u, v)χ (γ ,δ) ,Λ1 ,

1 vl∗ (x1 )eilx2 . PM,N,α,β,γ ,δ v(x) = |l|
N

vl∗

It can be checked that ∈ PM (Λ1 ) and al,α,β,γ ,δ (vl∗ − vl , φ) = 0, ∀φ ∈ PM (Λ1 ). Thus we use the above and (3.3) to verify that for any φ ∈ PM (Λ1 ),  ∗   2 v − vl 2 + l 2 vl∗ − vl χ (α2 ,β2 ) ,Λ l 1,α1 ,β1 ,γ ,δ,Λ1 1   = al,α,β,γ ,δ vl∗ − vl , vl∗ − vl     = al,α,β,γ ,δ vl∗ − vl , φ − vl
c φ − vl 21,α1 ,β1 ,γ ,δ,Λ1 + l 2 φ − vl 2χ (γ ,δ) ,Λ . 1

1 vl = PM,α 1 ,β1 ,γ ,δ,Λ1

Taking φ  ∗  v − vl 2 l 1,α

and using Lemma 2.4 and (3.3), we deduce that  2 + l 2 vl∗ − vl χ (α2 ,β2 ) ,Λ ,β ,γ ,δ,Λ 1 1 1 1  
cM 2−2r vl 2r,χ (α1 ,β1 ) ,∗,Λ + l 2 vl 2r+σ −2,χ (α1 ,β1 ) ,∗,Λ . 1

1

(3.6)

Next, let g ∈ L2χ (γ ,δ) (Λ1 ) and consider the following auxiliary problem: al,α,β,γ ,δ (w, z) = (g, z)χ (γ ,δ) ,Λ1 ,

∀z ∈ Hα11 ,β1 ,γ ,δ (Λ1 ).

(3.7)

Taking z = w in (3.7), we verify that w1,α1 ,β1 ,γ ,δ,Λ1 + l 2 wχ (α2 ,β2 ) ,Λ1
cgχ (γ ,δ) ,Λ1 .

(3.8)

Now let w(x1 ) vary in D(Λ1 ), and so by (3.7), in the sense of distributions,     −∂x1 ∂x1 w(x1 )χ (α1 ,β1 ) (x1 ) = g(x1 ) − w(x1 ) χ (γ ,δ) (x1 ) − l 2 w(x1 )χ (α2 ,β2 ) (x1 ).

(3.9)

As in the proof of Theorem 2.5 of [4], we can show that ∂x1 = ∂x1 w(−1) × χ (α1 ,β1 ) (−1) = 0. Moreover, we obtain from (3.9) that −1   −∂x21 w(x1 ) = − (α1 + β1 )x1 + (α1 − β1 ) 1 − x12 ∂x1 w(x1 )   + g(x1 ) − w(x1 ) χ (γ −α1 ,δ−β1 ) (x1 ) − l 2 w(x1 )χ (α2 −α1 ,β2 −β1 ) (x1 ). (3.10) w(1)χ (α1 ,β1 ) (1)

16

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28 (1)

(2)

We now estimate ∂x21 w(1 − x12 )1/2 χ (α1 ,β1 ) ,Λ1 . Let Λ1 = [0, 1) and Λ1 = (−1, 0). It can be verified that ∂x21 w(x1 )(1 − x12 )1/2 2 (α1 ,β1 )
D1 + D2 where ,Λ1

χ

 (1)   (2)  D1 = D1 Λ 1 + D1 Λ 1 ,

 (j )     2 ∂x1 w(x1 ) χ (α1 −1,β1 −1) (x1 ) dx1 , D1 Λ1 = 8 α12 + β12

j = 1, 2,

(j )

D2 = 2

Λ1

 2 g(x1 ) − w(x1 ) χ (2γ −α1 +1,2δ−β1 +1) (x1 ) dx1

Λ1

w 2 (x1 )χ (2α2 −α1 +1,2β2 −β1 +1) (x1 ) dx1 .

+ 2l 4 Λ1

Due to (3.3) and σ = 1, α1
min(γ + 1, α2 + 1) and β1
min(δ + 1, β2 + 1). So by (3.8),   |D2 |
c g − w2χ (γ ,δ) ,Λ + l 4 w2χ (α2 ,β2 ) ,Λ
cg2χ (γ ,δ) ,Λ . 1

1

1

(1)

Next, integrating (3.9) and inserting the result into D1 (Λ1 ), we obtain that for σ = 1,   D1 Λ(1) 1 1 2

   = c χ (−α1 −1,−β1 −1) (x1 ) g(y) − w(y) χ (γ ,δ) (y) − l 2 w(y)χ (α2 ,β2 ) (y) dy dx1 x1

(1)

Λ1

1
c

(1 − x1 )−γ

0

1 1 − x1

1

   g(y) − w(y) χ (γ ,δ) (y) − l 2 w(y)χ (α2 ,β2 ) (y) dy

2 dx1 .

x1

By the Hardy inequality, for any measurable function Φ(x1 ), real numbers a
b and d < 1, 2

b

b

b 1 4 d Φ(y) dy (b − x1 ) dx1
Φ 2 (x1 )(b − x1 )d dx1 . (3.11) b − x1 1−d a

x1

a

Letting a = −1, b = 1 and d = −γ in (3.11), we use (3.3) and (3.8) to derive that

 (1)   2 D1 Λ 1
c g(x1 ) − w(x1 ) χ (γ ,2δ) (x1 ) dx1 + l 4 w 2 (x1 )χ (2α2 −γ ,2β2 ) (x1 ) dx1 (1)

(1)

Λ1

Λ1


cg2χ (γ ,δ) ,Λ . 1

A similar estimate is valid on Λ(2) 1 for σ = 1. A combination of the previous statements leads to that for γ
α2 , δ
β2 and σ = 1,  1/2    (α ,β ) |w|2,χ (α1 ,β1 ) ,∗,Λ1 = ∂x21 w 1 − x12
cgχ (γ ,δ) ,Λ1 . (3.12) χ 1 1 ,Λ 1

vl∗

Now, we take z = − vl in (3.7), and use (3.3), (3.12) and the second result of Lemma 2.4 to obtain that for σ = 1 and |l|
N,

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

17

 ∗  v − vl , g (γ ,δ) l χ ,Λ1     1 = al,α,β,γ ,δ vl∗ − vl , w = al,α,β,γ ,δ vl∗ − vl , PM,α w−w 1 ,β1 ,γ ,δ,Λ1 2  2  1/2
vl∗ − vl 1,α ,β ,γ ,δ,Λ + l 2 vl∗ − vl χ (α2 ,β2 ) ,Λ 1 1 1 1 2 2  1 1/2  1 × PM,α w − w 1,α ,β ,γ ,δ,Λ + l 2 PM,α w − w χ (γ ,δ) ,Λ ,β ,γ ,δ,Λ 1 ,β1 ,γ ,δ,Λ1 1 1 1 1 1 1 1  2 1/2 −r 2 2
cM |vl |r,χ (α1 ,β1 ) ,∗,Λ + l |vl |r−1,χ (α1 ,β1 ) ,∗,Λ 1 1 1/2  2 2 −2 2 × |w|2,χ (α1 ,β1 ) ,∗,Λ + l M |w|2,χ (α1 ,β1 ) ,∗,Λ 1 1  1/2 
cM −r 1 + M −1 N |vl |2r,χ (α1 ,β1 ) ,∗,Λ + l 2 |vl |2r−1,χ (α1 ,β1 ) ,∗,Λ gχ (γ ,δ) ,Λ1 . 1

1

Consequently, vl∗ − vl χ (γ ,δ) ,Λ1 =

|(vl∗ − vl , g)χ (γ ,δ) ,Λ1 |

sup

gχ (γ ,δ) ,Λ1

g∈L2 (γ ,δ) (Λ1 ) χ

g =0


cM

−r

 1/2  1 + M −1 N vl 2r,χ (α1 ,β1 ) ,∗,Λ + l 2 vl 2r−1,χ (α1 ,β1 ) ,∗,Λ . 1

Therefore, 2  1  P M,N,α,β,γ ,δ v − v χ (γ ,δ) 

 ∗  v − vl 2 (γ ,δ) = 2π l

|l|
N

χ

,Λ1

+

|l|>N

(3.13)

1

 vl 2χ (γ ,δ) ,Λ

1

  2 
cM −2r 1 + M −1 N vl 2r,χ (α1 ,β1 ) ,∗,Λ + l 2 vl 2r−1,χ (α1 ,β1 ) ,∗,Λ + cN −2s

|l|>N

1

|l|
N

1

2   2 l 2s vl 2χ (γ ,δ) ,Λ
c M −r + N −s 1 + M −1 N v2

r,s,1 Mα,β,γ ,δ

1

.

The result (3.5) with 0
µ
1 comes from the previous results and space interpolation.

2

By using Lemmas 2.5 and 2.6, we can prove the following results. Theorem 3.5. Let γ
α2 and δ
β2 . If α1
γ + 1, β1
δ + 2, 0 < α1 < 1, β1 < 1, or r,s,2 1 α1
γ + 2, β1
0, δ  0, then for any v ∈ 0 Hα,β,γ ,δ (Ω) ∩ Mp,α,β,γ ,δ (Ω) and integers r, s  1,  1    1−r  0P  + N 1−s vM r,s,2 . M,N,α,β,γ ,δ v − v 1,α,β,γ ,δ
c M α,β,γ ,δ

If, in addition, β1 < δ + 1, then for all 0
µ
1,   1   0P M,N,α,β,γ ,δ v − v µ,α,β,γ ,δ µ  −r   1−µ M + N −s 1 + M −1 N
c M 1−r + N 1−s vM r,s,1 . α,β,γ ,δ

18

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

1 Theorem 3.6. Let γ
α2 and δ
β2 . If −1 < α1 , β1 < 1, then for any v ∈ H0,α,β,γ ,δ (Ω) ∩ r,s,2 Mp,α,β,γ ,δ (Ω) and integers r, s  1,   1,0   1−r  P˜ + N 1−s vM r,s,2 . M,N,α,β,γ ,δ v − v 1,α,β,γ ,δ
c M α,β,γ ,δ

If, in addition, α1 , β1
0 or α1 , β1 > 0, then for all 0
µ
1,  1,0  P˜ v − v M,N,α,β

µ,α,β,γ ,δ

µ  −r   1−µ M + N −s 1 + M −1 N
c M 1−r + N 1−s vM r,s,1 . α,β,γ ,δ

We now consider another orthogonal projection, which will be used for numerical solutions of semi-periodic problems on unbounded domains. Let ν = (ν0 , ν1 , ν2 ), νi > 0, and  α1 β1  α1 β1    ν aˆ α,β,0,0 (u, v) = ν1 χ ( 2 , 2 ) ∂x1 u, ∂x1 χ ( 2 , 2 ) v + ν2 χ (α2 ,β2 ) ∂x2 u, ∂x2 v + ν0 (u, v). Lemma 3.1. Let α2 , β2 > −1. If α1 = β1 = 2, ν0 > 2ν1 , or α1 = 2, β1 = 0, ν0 > 12 ν1 , then for 1 any u, v ∈ Hp,α,β,0,0 (Ω), ν aˆ α,β,0,0 (v, v)  c(ν)v21,α,β,0,0 ,

(3.14)

ν aˆ α,β,0,0 (u, v)
cu1,α,β,0,0 v1,α,β,0,0 ,

(3.15)

where c(ν) = min(ν0 − 2ν1 , ν1 , ν2 ) for ν0 > 2ν1 , and c(ν) = min(ν0 − 12 ν1 , ν1 , ν2 ) for ν0 > 12 ν1 . Proof. Let vl (x1 ) be the same as in (3.2). If α1 = β1 = 2, then integrating by parts yields that  α1 β1   ( α1 , β1 ) χ 2 2 ∂x1 vl , ∂x1 χ ( 2 , 2 ) vl Λ 1

    = ∂x1 vl 2χ (α1 ,β1 ) ,Λ − x1 1 − x12 ∂x1 vl2 (x1 ) dx1 1

Λ1

= ∂x1 vl 2χ (α1 ,β1 ) ,Λ 1

+ vl 2Λ1

−3

 ∂x1 vl 2χ (α1 ,β1 ) ,Λ 1

− 2vl 2Λ1 .

x12 vl2 (x1 ) dx1

Λ1

Thus ν aˆ α,β,0,0 (v, v)

= 2π

∞

  ( α1 , β1 )  α1 β1   ν1 χ 2 2 ∂x1 vl , ∂x1 χ ( 2 , 2 ) vl Λ + ν2 l 2 vl 2χ (α2 ,β2 ) ,Λ + ν0 vl 2Λ1 1

|l|=0

1

∞

   2π ν1 ∂x1 vl 2χ (α1 ,β1 ) ,Λ + ν2 l 2 vl 2χ (α2 ,β2 ) ,Λ + (ν0 − 2ν1 )vl 2Λ1 |l|=0

1

 min(ν0 − 2ν1 , ν1 , ν2 )v21,α,β,0,0 .

1

(3.16)

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

19

The above implies the result (3.14) with ν0 > 2ν1 . If α1 = 2 and β1 = 0, then  α1 β1   ( α1 , β1 ) 1 χ 2 2 ∂x1 vl , ∂x1 χ ( 2 , 2 ) vl Λ = ∂x1 vl 2χ (α1 ,β1 ) ,Λ − vl 2Λ1 + vl2 (−1). 1 1 2 This fact leads to (3.14) with ν0 > 12 ν1 . The proof of (3.15) is clear. 2 1 1 The orthogonal projection PˆM,N,α,β,0,0 : Hp,α,β,0,0 (Ω) → VM,N is defined by   ν 1 aˆ α,β,0,0 PˆM,N,α,β,0,0 v − v, φ = 0, ∀φ ∈ VM,N . 1 1 : 0 Hp,α,β,0,0 (Ω) → 0 VM,N is defined by The orthogonal projection 0 PˆM,N,α,β,0,0   ν 1 aˆ α,β,0,0 0 PˆM,N,α,β,0,0 v − v, φ = 0, ∀φ ∈ 0 VM,N .

For description of approximation result, we introduce the space    r,s Mˆ α,β,,0,0 (Ω) = L2 Λ2 ; H r α1 β1 (Λ1 ) ∩ H 1 Λ2 ; H r−1 α1 β1 χ

( 2 , 2 ) ,∗

χ

( 2 , 2 ) ,∗

 (Λ1 )

    ∩ H s−1 Λ2 ; Hα11 ,β1 ,0,0 (Λ1 ) ∩ H s Λ2 ; L2χ (α2 ,β2 ) (Λ1 ) . Its norm vMˆ r,s

α,β,0,0

is defined in the usual way.

Theorem 3.7. Let α2 , β2  0, and the conditions of Lemma 3.1 hold. Then for any v ∈ r,s (Ω) and integers r, s  1, Mˆ p,α,β,0,0  1    1−r Pˆ  + N 1−s vMˆ r,s . M,N,α,β,0,0 v − v 1,α,β,0,0
c M α,β,0,0

Proof. Let vl (x1 ) be the same as before, and

  P 1 α1 β1 v ∗ (x) = vl (x1 ) eilx2 . |l|
N

M,

2

,

2

,0,0,Λ1

1 Clearly, v ∗ ∈ VM,N . So by Lemma 3.1 and the definition of PˆM,N,α,β,0,0 ,  1 2 c(ν)PˆM,N,α,β,0,0 v − v 1,α,β,0,0  1  ν 1 PˆM,N,α,β,0,0 v − v, PˆM,N,α,β,0,0
aˆ α,β,0,0 v−v  1   1  ν = aˆ α,β,0,0 PˆM,N,α,β,0,0 v − v, φ − v
cPˆM,N,α,β,0,0 v − v 1,α,β,0,0 φ − v1,α,β,0,0 .

Taking φ = v ∗ and using Lemma 2.4, we verify that for r  2 and s  1,  1 2 Pˆ  M,N,α,β,0,0 v − v 1,α,β,0,0 

 1 2  P α β
c vl − vl 1, α1 , β1 ,0,0,Λ + l 2 P 1 α1 β1 1 1 |l|
N

+c

M,

2

,

2

,0,0,Λ1

2

1

2

M,

2

,

,0,0,Λ1

2

2 vl − vl 

Λ1

  vl 21,α1 ,β1 ,0,0,Λ1 + l 2 vl 2χ (α2 ,β2 ) ,Λ 1

|l|>N


cM 2−2r v2L2 (Λ

2

;H r

(Λ1 α β ( 1, 1) χ 2 2 ,∗

 + cN 2−2s v2H s−1 (Λ

+ v2 ))

1 2 ;Hα ,β ,0,0 (Λ1 )) 1 1

H 1 (Λ

 r−1 (Λ1 )) 2 ;H α β ( 1, 1) χ 2 2 ,∗

+ v2H s (Λ



2 2 ;L (α ,β ) (Λ1 )) χ 2 2

.

2



20

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

In numerical analysis of the mixed Fourier–Jacobi spectral methods for nonlinear problems, 1 vL∞ . we have to estimate the norm PˆM,N,α,β,0,0 Theorem 3.8. If M = O(N ), and the conditions of Theorem 3.7 hold, then for any v ∈ (Λ1 ) ∩ H d (Λ1 )) and d > 1, L2p (Λ2 ; H 1+d α1 β1 χ

( 2 , 2 ) ,∗

  1  Pˆ M,N,α,β,0,0 v

L∞


cvL2 (Λ2 ;H 1+d

(Λ1 )∩H α β ( 1, 1) χ 2 2 ,∗

d (Λ

1 ))

.

Proof. Let

 α1 β1  α1 β1  ν aˆ l,α,β,0,0 (u, v) = ν1 χ ( 2 , 2 ) ∂x1 u, ∂x1 χ ( 2 , 2 ) v Λ + ν2 l 2 (u, v)χ (α2 ,β2 ) ,Λ1 + ν0 (u, v)Λ1 , 1

1 ∗ ilx2 ˆ vl (x1 )e ∈ PM (Λ1 ). PM,N,α,β,0,0 v(x) = |l|
N

By an argument as in the proof of Lemma 3.1, we assert that for certain c(ν) > 0,   ν (w, w)  c(ν) w21,α1 ,β1 ,0,0,Λ1 + l 2 w2χ (α2 ,β2 ) ,Λ , aˆ l,α,β,0,0 1 ν aˆ (w, z) l,α,β,0,0  1/2  2 1/2
c w21,α1 ,β1 ,0,0,Λ1 + l 2 w2χ (α2 ,β2 ) ,Λ z1,α1 ,β1 ,0,0,Λ1 + l 2 z2χ (α2 ,β2 ) ,Λ ) . 1

1

ν (vl − vl∗ , φ) = 0 for all φ ∈ PM (Λ1 ). Hence Moreover, aˆ l,α,β,0,0     2 ∗ 2 vl − v ∗ 2 l 1,α1 ,β1 ,0,0,Λ1 + l vl − vl χ (α2 ,β2 ) ,Λ1     ν ν vl − vl∗ , vl − vl∗ = caˆ l,α,β,0,0 vl − vl∗ , vl − φ
caˆ l,α,β,0,0  
c φ − vl 21,α1 ,β1 ,0,0,Λ1 + l 2 φ − vl 2χ (α2 ,β2 ) ,Λ . 1

By taking φ

vl = P 1 α1 β1 M, 2 , 2 ,0,0,Λ1

  vl − v ∗ 2

and using Lemma 2.4, we deduce that for |l|
N, 



2 + l 2 vl − vl∗ χ (α2 ,β2 ) ,Λ 1,α1 ,β1 ,0,0,Λ1 1  
cM 2−2r 1 + N 2 M −2 vl 2 α1 β1 . , ) ( r,χ 2 2 ,∗,Λ1 l

(3.17)

We now use (3.17) to derive the desired result. Due to d > 1 and the imbedding theorem,  ∗   ∗   v  ∞ l L ,Λ1
vl L∞ ,Λ1 + c vl − vl d2 ,Λ1    ∗
vl L∞ ,Λ1 + c vl − PM,0,0,Λ1 vl  d ,Λ + PM,0,0,Λ1 vl − vl  d ,Λ1 . 2

1

2

Moreover, by Lemmas 2.2, 2.3, (3.17) and the fact M = O(N ),  ∗     v − PM,0,0,Λ vl  d
cM d vl∗ − vl Λ + PM,0,0,Λ1 vl − vl Λ1 1 l 1 2 ,Λ1   + v 
c vl  α1 β1 l d,Λ1 . , ) ( 1+d,χ

2

2

,∗,Λ1

On the other hand (see [2]), PM,0,0,Λ1 vl − vl  d ,Λ1
cvl  3d ,Λ1 . A combination of the previous 2 4 estimates implies that    ∗ v  ∞
c v  + v  β α l l d,Λ 1 1 1 . l L ,Λ , ) ( 1

1+d,χ

2

2

,∗,Λ1

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

Finally, the above with the result on the Fourier approximation lead to

PˆM,N,α,β,0,0 v2L∞
c vl∗ 2L∞ ,Λ1
cv2 2 1+d L (Λ2 ;H

|l|
N

α β ( 1, 1) χ 2 2 ,∗

21

(Λ1 )∩H d (Λ1 ))

.

2

Remark 3.1. For α2 = β2 = 0, we can remove the condition M = O(N ) in Theorem 3.6. Indeed, . So the conclusion follows as in the late part in this case, vl∗ − vl Λ1
cM 1−r vl  ( α1 , β1 ) r,χ

2

2

of the proof of Theorem 3.8.

,∗,Λ1

By using Lemmas 2.5, 3.1 and the same argument as in the proof of Theorem 3.7, we can prove following result. 1 Theorem 3.9. Let α1 = 2, β1 = 0, α2  2, β2 = 0 and ν0 > 12 ν1 . Then for any v ∈0 Hα,β,0,0 (Ω) ∩ r,s ˆ Mp,α,β,0,0 (Ω) and integers r, s  1,   1   1−r   0 Pˆ + N 1−s vMˆ r,s . M,N,α,β,0,0 v − v 1,α,β,0,0
c M α,β,0,0

4. Some applications We first consider the following simple model problem,     −∂x1 a1 (x)∂x1 U (x) − ∂x2 a2 (x)∂x2 U (x) + a0 (x)U (x) = f (x),

x ∈ Ω.

(4.1)

Assume that ai (x) degenerate as |x1 | → 1, and a0 (x) = a˜ 0 (x)χ (γ ,δ) (x),

a1 (x) = a˜ 1 (x)χ (α1 ,β1 ) (x),

a2 (x) = a˜ 2 (x)χ (α2 ,β2 ) (x), with a˜ i

(x) ∈ L∞ (Ω),

(4.2)

(i) a˜ i (x)  a˜ min

> 0, i = 0, 1, 2. We look for solution of (4.1) such that

lim a1 (x)U (x)∂x1 U (x) = 0,

|x1 |→1

∀x2 ∈ Λ2 .

(4.3)

1 For any u, v ∈ Hα,β,γ ,δ (Ω), let

Aα,β,γ ,δ (u, v) = (a˜ 1 ∂x1 u, ∂x1 v)χ (α1 ,β1 ) + (a˜ 2 ∂x2 u, ∂x2 v)χ (α2 ,β2 ) + (a˜ 0 u, v)χ (γ ,δ) . 1 A weak formulation of (4.1) with (4.3) is to find U ∈ Hp,α,β,γ ,δ (Ω) such that

Aα,β,γ ,δ (U, v) = (f, v),

1 ∀v ∈ Hp,α,β,γ ,δ (Ω).

1 By (4.2) and the Cauchy–Schwartz inequality, we find that for any u, v ∈ Hp,α,β,γ ,δ (Ω),   Aα,β,γ ,δ (u, v)
a˜ 1 L∞ + a˜ 2 L∞ + c˜0 L∞ u1,α,β,γ ,δ v1,α,β,γ ,δ ,   Aα,β,γ ,δ (u, u)  min a˜ (0) , a˜ (1) , a˜ (2) u2 1,α,β,γ ,δ . min min min

(4.4)

(4.5) (4.6)

Therefore, if f ∈ L2χ (−γ ,−δ) (Ω), then (4.4) has a unique solution such that U 1,α,β,γ ,δ
cf χ (−γ ,−δ) . Now, let uM,N ∈ VM,N be the approximation to U , satisfying Aα,β,γ ,δ (uM,N , φ) = (f, φ),

∀φ ∈ VM,N .

Due to (4.2), (4.7) is also unisolvent and uM,N 1,α,β,γ ,δ
cf χ (−γ ,−δ) .

(4.7)

22

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

Theorem 4.1. Let U and uM,N be the solutions of (4.1) and (4.7), respectively. If (3.3) and (4.2) r,s,σ hold, and U ∈ Mp,α,β,γ ,δ (Ω) with integers r, s  1 and σ = 1, 2, then   U − uM,N 1,α,β,γ ,δ
d M 1−r + N 1−s U M r,s,σ , α,β,γ ,δ

where the constant d depends only on the norms a˜ i L∞ , i = 0, 1, 2. 1 Proof. Let UM,N = PM,N,α,β,γ ,δ U. By (4.5)–(4.7) and Theorem 3.4,

cuM,N − UM,N 21,α,β,γ ,δ
Aα,β,γ ,δ (uM,N − UM,N , uM,N − UM,N ) = Aα,β,γ ,δ (U − UM,N , uN − UM,N )  
d N11−r + N21−s U M r,s,σ uM,N − UM,N 1,α,β,γ ,δ . α,β,γ ,δ

Thus using Theorem 3.4 again completes the proof.

2

We next consider the nonlinear Klein–Gordon equation, which plays an important role in quantum mechanics. Let Ω ∗ = {y = (y1 , y2 ) | −∞ < y1 < +∞, 0
y2 < 2π}. The problem is of the form  2 3 ∗   ∂t V (y, t) − ∆V (y, t) + V (y, t) = F (y, t), (y, t) ∈ Ω × (0, T ], (4.8) ∂t V (y, 0) = V1 (y, 0), y ∈ Ω ∗,   V (y, 0) = V1 (y, 0), y ∈ Ω ∗. Assume that all functions in (4.8) have the period 2π for the variable y2 , and V (y, t) satisfies the following asymptotic boundary condition: lim e−a|y1 | ∂y1 V (y1 , y2 , t) = 0,

|y1 |→∞

a > 0, ∀(y2 , t) ∈ Λ2 × [0, T ].

(4.9)

Let b > 0, which should be chosen suitably based on the value of a. We make the following variable transformation: 1 + x1 y1 (x1 ) = b ln , y2 (x2 ) = x2 . (4.10) 1 − x1 Furthermore,   U (x1 , x2 , t) = V y1 (x1 ), y2 (x2 ), t ,   U0 (x1 , x2 ) = V0 y1 (x1 ), y2 (x2 ) ,

  U1 (x1 , x2 ) = V1 y1 (x1 ), y2 (x2 ) ,   f (x1 , x2 , t) = F y1 (x1 ), y2 (x2 ), t .

Then problem (4.8) with (4.9) becomes  2 ∂t U (x, t) − 4b12 (1 − x12 )∂x1 ((1 − x12 )∂x1 U (x, t)) − ∂x22 U (x, t) + U 3 (x, t) = f (x, t),     (x, t) ∈ Ω × (0, T ], (4.11) 2 ab+1 ∂ U (x, t) = 0, (x , t) ∈ Λ × [0, T ],   x1 2 2  lim|x1 |→1 (1 − x1 )  ∂t U (x, 0) = U1 (x), U (x, 0) = U0 (x), x ∈ Ω. We now derive a weak form of (4.11). Multiplying the second term of (4.11) by v ∈ 1 (Ω) ∩ L4p (Ω) and integrating the result by parts, we obtain the boundary term Hp,α,β,0,0 B(U, v) =

 2  2 1 1 lim 1 − x12 v(x)∂x1 U (x) − 2 lim 1 − x12 v(x)∂x1 U (x). 2 4b x1 →−1 4b x1 →1

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23

Since v ∈ L4 (Ω), we have v = o((1 − x12 )−1/4 ) as |x1 | → 1. By the boundary condition (4.3), B(U, v) = 0 for ab
34 . Next, let α1 = β1 = 2, α2 = β2 = 0, ν0 > 2b12 , ν1 = 4b12 and 1 (Ω)) ∩ ν2 = 1. Then the weak formulation of (4.11) is to find U ∈ L∞ (0, T ; L4p (Ω) ∩ Hp,α,β,0,0 1,∞ 2 (0, T ; Lp (Ω)) such that W  2 ν 3   (∂t U (t) + U (t), v) + aˆ α,β (U (t), v) = (ν0 U (t) + f (t), v), 1 (4.12) ∀v ∈ L4p (Ω) ∩ Hp,α,β,0,0 (Ω), t ∈ (0, T ],   ∂t U (0) = U1 , U (0) = U0 . Obviously the condition of Lemma 3.1 is fulfilled. So for suitable U1 , U0 and f, (4.12) has a unique solution. The mixed Fourier–Jacobi spectral scheme for (4.12) is to find uM,N (t) ∈ VM,N such that  2 3 ν   (∂t uM,N (t) + uM,N (t), φ) + aˆ α,β,0,0 (uM,N (t), φ) = (ν0 uM,N (t) + f (t), φ),   ∀φ ∈ V M,N , t ∈ (0, T ], (4.13)  ∂t uM,N (0) = uM,N,1 = PM,N,0,0 U1 ,    uM,N (0) = uM,N,0 = PM,N,0,0 U0 . We now analyze the stability of scheme (4.13). Assume that the data uM,N,0 , uM,N,1 and f are disturbed by u˜ M,N,0 , u˜ M,N,1 and f˜, respectively, which induce the error of uM,N , denoted by u˜ M,N . Then for all φ ∈ VM,N and t ∈ (0, T ],       2 ν u˜ M,N (t), φ = ν0 u˜ M,N (t) + f˜(t), φ , (4.14) ∂t u˜ M,N (t) + u˜ 3M,N (t) + G0 (t), φ + aˆ α,β with G0 (t) = 3u˜ 2M,N (t)uM,N (t) + 3u˜ M,N (t)u2M,N (t), ∂t u˜ M,N (0) = u˜ M,N,1 ,

u˜ M,N (0) = u˜ M,N,0 .

Let 2 4  2 1 1 E(u˜ M,N , t) = u˜ M,N (t)1,α,β,0,0 + u˜ M,N (t)L4 + ∂t u˜ M,N (t) . 4 2 We take φ = 2∂t u˜ M,N (t) in (4.14). Thanks to Lemma 3.1 and the fact that       2 G0 (t), ∂t u˜ M,N (t)
1 u˜ M,N (t)4 4 + 1 u˜ M,N (t)2 L 2 2 2  4  2    + 18 uM,N (t) ∞ + uM,N (t)∞ ∂t u˜ M,N (t) , we obtain

2  ∂t E(u˜ M,N , t)
d(uM,N )E(u˜ M,N , t) + f˜(t) ,

(4.15)

where d(uM,N ) is a positive constant depending only on uM,N L∞ (0,T ;L∞ (Ω)) . Let ρ(u˜ M,N,0 , u˜ M,N,1 , f˜, t) 1 1 = u˜ M,N,0 21,α,β,0,0 + u˜ M,N,0 4L4 + u˜ M,N,1 2 + 4 2

t

  f˜(s)2 ds.

0

Integrating (4.15) with respect to t, we obtain the following result.

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L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

Theorem 4.2. Let uM,N be the solution of (4.13), and u˜ M,N be its error induced by u˜ M,N,0 , u˜ M,N,1 and f˜. If ab
34 , then for all 0
t
T , E(u˜ M,N , t)
ρ(u˜ N,0 , u˜ N,1 , f˜, t)ed(uM,N )t . We next deal with the convergence of the scheme (4.13). To obtain better error estimate, we 1 compare the numerical solution uM,N with UM,N = PˆM,N,α,β,0,0 U. By (4.12), 3    

  2 3 ν UM,N (t), φ + Gj (t), φ (t), φ + aˆ α,β,0,0 ∂t UM,N (t) + UM,N j =1

  = ν0 UM,N (t) + f (t), φ ,

∀φ ∈ VM,N , t ∈ (0, T ],

(4.16)

with G1 (t, φ) = ∂t2 U (t) − ∂t2 UM,N (t), G3 (t, φ) = U

3

G2 (t, φ) = ν0 UM,N (t) − ν0 U (t),

3 (t) − UM,N (t).

Furthermore, let U˜ M,N = uM,N − UM,N . Then subtracting (4.16) from (4.13) gives that  2    3 ˜ ˜ 0 (t), φ + aˆ ν ∂t U˜ M,N (t) + U˜ M,N (t) + G α,β,0,0 UM,N (t), φ 3  

  Gj (t), φ , = ν0 U˜ M,N (t), φ +

∀φ ∈ VM,N , t ∈ (0, T ],

(4.17)

j =1

with 2 2 ˜ 0 (t) = 3U˜ M,N (t)UM,N (t) + 3U˜ M,N (t)UM,N (t), G 1 U1 , ∂t U˜ M,N (0) = PM,N,0,0 U1 − PˆM,N,α,β,0,0 1 U0 . U˜ M,N (0) = PM,N,0,0 U0 − PˆM,N,α,β,0,0

Comparing (4.14) with (4.17), we build up an error estimation similar to (4.15). But uM,N , u˜ M,N , u˜ M,N,0 , u˜ M,N,1 and uM,N L∞ (0,T ;L∞ (Ω)) in (4.15) are now replaced by UM,N , U˜ M,N , U˜ M,N (0), ∂t U˜ M,N (0) and UM,N L∞ (0,T ;L∞ (Ω)) , respectively. Thus it remains to estimate Gj (t), ∂t U˜ M,N (0), U˜ M,N (0)1,α,β,0,0 and U˜ M,N (0)L4 . Firstly, we have from Theorem 3.7 that       G1 (t) + G2 (t) = ∂ 2 UM,N (t) − ∂ 2 U (t) t t      
c M 1−r + N 1−s ∂t2 U (t)Mˆ r,s + U (t)Mˆ r,s . α,β,0,0

α,β,0,0

Next, by Remark 3.1, the imbedding theorem and Theorems 3.7 and 3.8, we get that for d > 1,         G3 (t)
c UM,N (t)2 + U (t)2 UM,N (t) − U (t) ∞ ∞   
cM(U ) M 1−r + N 1−s U (t)Mˆ r,s , α,β,0,0

where

 2 M(U ) = max U (t)L2 (Λ 0
t
T

1+d 2 ;H α χ

β ( 21 , 21 ) ,∗

(Λ1 )∩H d (Λ1 ))

 2  + U (t)H d (Ω) .

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25

r,s ˆ r,s For simplicity, let integers r, s  1 and Zα,β (Ω) = Hχr−1,s−1 (0,0) ,A (Ω) ∩ Mα,β,0,0 (Ω). Its norm vZ r,s is defined in the usual way. Then by Theorems 3.3 and 3.7, α,β

    ∂t U˜ M,N (0)
PM,N,0,0,0,0 U1 − U1  + Pˆ 1  M,N,α,β,0,0 U1 − U1  1−r 
c M + N 1−s U1 Z r,s . α,β

Since α1 = β1 = 2 and α2 = β2 = 0, we use Theorems 3.2, 3.3 and 3.7 to derive that   2  2 2   U˜ M,N (0)2
U˜ M,N (0) 1,1 + U˜ M,N (0)
c(M + N )2 U˜ M,N (0) 1,α,β,0,0

H

χ (α,β)

 2
c(M + N )2 M −r + N −s U0 2 r+1,s+1 . Zα,β

Moreover, by Theorems 3.1, 3.3 and 3.7,           U˜ M,N (0) 4
c M 2 N 1/4 U˜ M,N (0)
c M 2 N 1/4 M −r + N −s U0  r+1,s+1 . L Z α,β

A combination of the previous estimates leads to the following result. Theorem 4.3. Let U and uM,N be the solutions of (4.12) and (4.13), respectively. Assume that for α1 = β1 = 2, α2 = β2 = 0, d > 1 and integers r, s  1,      r,s d U ∈ L∞ 0, T ; Hpd (Ω) ∩ L2p Λ2 ; H 1+d (Λ ) ∩ H (Λ ) ∩ H 2 0, T ; Mˆ p,α,β,0,0 (Ω) , 1 1 α1 β1 χ

r+1,s+1 U0 ∈ Zp,α,β (Ω),

( 2 , 2 ) ,∗

r,s U1 ∈ Zp,α,β (Ω).

Then for all 0
t
T ,  2  2 E(uM,N − U, t)
b∗ M 1−r + N 1−s + (M + N )2 M −r + N −s 4    + M 2 N M −r + N −s , b∗ being a positive constant depending on the norms of U, U0 , U1 and f in the mentioned spaces. Remark 4.1. If M = O(N ), then the result in Theorem 4.3 is reduced to E(uM,N − U, t)
1 b∗ (M 1−r + N 1−s )2 . On the other hand, if we take uM,N (0) = PˆM,N,0,0,0,0 U0 and ∂t uM,N (t) = 1 ˆ PM,N,0,0,0,0 U1 , then we can weaken the conditions on U0 and U1 in Theorem 4.3, and obtain the same result. The mixed Fourier–Jacobi approximation is also applicable to other problems. For instance, we consider the heat transfer inside a unit disc,  1 1 2   ∂t V − r ∂r (r∂r V ) − r 2 ∂θ V = F, (r, θ, t) ∈ (0, 1) × [0, 2π) × (0, T ], (4.18) V (1, θ, t) = 0, (θ, t) ∈ [0, 2π) × [0, T ],   V (r, θ, 0) = V0 (r, θ ), (r, θ ) ∈ [0, 1] × [0, 2π), where all functions have the period 2π for the variable θ. In addition, the solution V satisfies the 1 polar condition: ∂θ V (0, θ, t) = 0. Let r = 1−x 2 , θ = x2 , x = (x1 , x2 ), η(x) = 1 − x1 and

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 1 − x1 , x2 , t , U (x1 , x2 , t) = V 2   1 − x1 f (x1 , x2 , t) = F , x2 , t . 2





U0 (x1 , x2 ) = V0

 1 − x1 , x2 , 2

Then problem (4.18) reads  4 2   η∂t U − 4∂x1 (η∂x1 U ) − η ∂x2 U = ηf, (x, t) ∈ Ω × (0, T ], U (−1, x2 , t) = ∂x2 U (1, x2 , t) = 0, (x2 , t) ∈ Λ2 × [0, T ],   U (x, 0) = U0 (x), x ∈ Ω.

(4.19)

The mixed Fourier–Jacobi approximation can also be used for some exterior problems. As an example, we consider the Laplace equation outside a unit disc,  1 (r, θ ) ∈ (1, +∞) × [0, 2π), − r ∂r (r∂r V ) − r12 ∂θ2 V = F, (4.20) a V (1, θ ) = limr→∞ r ∂r V (r, θ ) = 0, a < 2, θ ∈ [0, 2π). Let

 2 θ = x2 , U (x1 , x2 ) = V , x2 , 1 − x1   4 2 f (x1 , x2 ) = F , x2 . 3 1 − x1 (1 − x1 ) 

2 r= , 1 − x1

Then (4.20) is changed into
−∂x1 (η∂x1 U ) − η∂x22 U = f, U (−1, x2 ) = limx1 →1 η2−a ∂x1 U (x1 , x2 ) = 0,

(x1 , x2 ) ∈ Ω, x2 ∈ Λ 2 .

(4.21)

Clearly we may use the Fourier–Jacobi spectral method to solve (4.19) and (4.21). In the end of this paper, we present some numerical results. We first consider problem (4.1) with a1 (x) = a2 (x) = 1 − x12 and a0 (x) = 1, and take the test function U (x) = 1 arcsin(x1 ) sin( 10 x1 + 2x2 ). Clearly, |∂x1 U | → ∞ as |x1 | → 1. We use scheme (4.7) with αi = βi = 1, i = 1, 2, and γ = δ = 0, to solve (4.1) numerically. For description of the errors, let j j Λ1,M = {x1 | 0
j
M} be the set of Gauss–Legendre nodes, Λ2,N = {x2 = πj N | 0
j
2N }, and U (x) − uM,N (x) , Eabs (uM,N ) = max x∈Λ1,M ×Λ2,N

U (x) − uM,N (x) . Erel (uM,N ) = max x∈Λ1,M ×Λ2,N U (x) We plot in Fig. 1 the errors with different M and N, which show the convergence of (4.7). We next consider problem (4.8) and (4.9) with a = 12 , and take f in such a way that (4.12) has 1 1 the exact solution U (x, t) = (1 − x)γ (1 + x)δ exp( 10 sin(x1 + 2x2 ) + 10 t). Its regularity depends on the values of γ and δ, essentially. We use scheme (4.13) with α1 = β1 = 2, α2 = β2 = 0, ν0 = 1, ν1 = 14 and ν2 = 1, to solve (4.12) numerically. In actual computation, we need to discretize (4.13) in time t. To do this, let τ be the mesh size in time, Rτ = {t = kτ | k = 1, 2, . . . , [ Tτ ]}, and

L.L. Wang, B.Y. Guo / J. Math. Anal. Appl. 315 (2006) 8–28

27

Fig. 1. The errors with different M = N.

Fig. 2. The errors with various M = N.

 1 v(x, t + τ ) + v(x, t − τ ) , 2  1 Dˆ τ τ v(x, t) = 2 v(x, t + τ ) − 2v(x, t) + v(x, t − τ ) . τ To increase the stability of computation, we approximate the nonlinear term in (4.12) by v(x, ˆ t) =

3

  ˆ v(x, t) = 1 v j (x, t + τ )v 3−j (x, t − τ ). G 4 j =0

The fully discrete scheme is         ν ˆ uM,N (t) , φ + aˆ α,β uˆ M,N (t), φ = ν0 uˆ M,N (t) + fˆ(t), φ , Dˆ τ τ uM,N (t) + G ∀φ ∈ VM,N , t ∈ Rτ ,

(4.22)

with uM,N (0) = PM,N,0,0 U0 ,

uM,N (τ ) = PM,N,0,0 U0 + τ PM,N,0,0 U1 .

Now, let y1 (x1 ) and y2 (x2 ) be the same as in (4.10) with b = 1. The numerical solution of the y original problem is vM,N (y1 , y2 , t) = uM,N ( eey11 −1 +1 , y2 , t). Let

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Fig. 3. The errors with various t .

 j j  j  Λ∗1,M = y1 = y1 x1 x1 ∈ Λ1,M , 0
j
M ,  j j  j  Λ∗2,N = y2 = y2 x2 x2 ∈ Λ2,M , 0
j
2N . Denote by Eabs (vM,N , t) and Erel (vM,N , t) the absolute error and the relative error of V (y, t) − vM,N (y, t) on the grid set Λ∗1,M × Λ∗2,N at time t. We plot in Fig. 2 the errors with γ = δ = 10−2 , τ = 10−3 and t = 1, which show the convergence of scheme (4.22). In Fig. 3, we plot the errors with γ = δ = 10−2 , τ = 10−3 and M = N = 32 in different time t, which demonstrate the stability of computation. References [1] C. Bernardi, M. Dauge, Y. Maday, in: P.G. Ciarlet, P.L. Lions (Eds.), Spectral Methods for Axisymmetric Domains, in: Ser. Appl. Math., vol. 3, Gauthier–Villars, North-Holland, Paris, 1999. [2] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988. [3] B.Y. Guo, Gegenbauer approximation in certain Hilbert spaces and its applications to singular differential equations, SIAM J. Numer. Anal. 37 (2000) 621–645. [4] B.Y. Guo, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. Math. Anal. Appl. 243 (2000) 373–408. [5] B.Y. Guo, H.P. Ma, W.M. Cao, H. Huang, The Fourier–Chebyshev spectral method for solving two-dimensional unsteady vorticity equations, J. Comput. Phys. 101 (1992) 207–217. [6] B.Y. Guo, L.L. Wang, Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math. 14 (2001) 227–276. [7] B.Y. Guo, L.L. Wang, Non-isotropic Jacobi spectral method, in: Contemp. Math., vol. 329, 2003, pp. 157–164. [8] B.Y. Guo, Y.S. Xiong, A spectral-difference method for two-dimensional viscous flow, J. Comput. Phys. 84 (1989) 259–278. [9] D.B. Ingham, Flow past a suddenly heated vertical plate, Proc. Roy. Soc. London Ser. A 402 (1985) 109–134. [10] M.G. Macaraeg, A combined spectral-finite element method for solving unsteady Navier–Stokes equations, J. Comput. Phys. 101 (1992) 375–385. [11] P. Mion, J. Kim, Numerical investigation of turbulent channel flow, J. Fluid Mech. 118 (1982) 341–377. [12] J.W. Murdok, Three-dimensional numerical study of boundary stability, AIAA 86-0434 (1986).