National Sun Yat-sen University Institutional Repository:Item 987654321/39899

行政院國家科學委員會專題研究計畫 成果報告

Trefftz 與邊界方法

計畫類別： 個別型計畫 計畫編號： NSC94-2115-M-110-013- 執行期間： 94 年 08 月 01 日至 95 年 07 月 31 日 執行單位： 國立中山大學應用數學系(所) 計畫主持人： 李子才 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢

中 華 民 國 95 年 10 月 13 日

行政院國家科學委員會專題研究計畫期末報告

Trefftz 與邊界方法

計畫編號：94-2115-M-110-013

執行期限：94 年 8 月 1 日至 95 年 07 月 31 日

主持人：李子才 國立中山大學應用數學系

一、中文摘要 本計劃旨在研究 Trefftz 法(即邊界逼 近 法 ) 用 於 Laplace 特 徵 值 問 題 。 對 解 Laplace 特徵值問題，我們使用 Trefftz 法發 展新的算法。因為分片特解能被完全使 用，新的算法有利於特徵值與特徵函數的 高精度，降低了 CPU 的時間與記憶體空 間。這算法也能應用去解多介面與奇異問 題。在這論文中特徵值與特徵函數的逼近 誤差分析被導出。平滑解與奇異解的數值 驗算被報告去驗證程序的有效性與理論分 析的結果。此文(即[21])已被 J. Comp. and Applied Math.錄取。 關鍵詞： Trefftz 法， 特徵值問題，邊界 逼近法，Helmholtz 方程，誤差分析。 Abstract

This research plan aims at developments of Trefftz method for Laplace's eigenvalue problems. For solving Laplace's eigenvalue problems we propose new algorithms using the Trefftz method (TM) (i,e., the boundary approximation method (BAM)), by means of degeneracy of numerical Helmholtz equations. Since piecewise particular solutions can be fully adopted, the new algorithms benefit high accuracy of eigenvalues and eigenfunctions, low cost in CPU time and computer storage. Also the algorithms can be applied to solve the problems with multiple interfaces and singularities. In this paper, error estimates are derived for the approximate eigenvalues and eigenfunctions obtained. Numerical experiments for smooth and singular solutions are reported in this paper to show significance of the algorithms proposed and to verify the theoretical results made.

Keywords: Trefftz method, eigenvalue

problems, boundary approximation method,

Helmholtz equation, error analysis. 二、緣由與目的

使用特解技巧解特徵值問題已有 Bergman [1]，Eisenstat [2]，Fox, Henrici & Moler [3]，Mathon & Sermer [4] 與 Vekua [5]。在 這論文中我們沿用[3]的觀念與採用 Li et al. [6]的分片特解構造迭代算法。我們將解非 齊邊界條件的輔助 Helmholtz 方程而且選 擇參數 去逼近特徵值。對解的區域被分 成一些子區域而在不同的子區域使用不同 的 Helmholtz 方程特解。Helmholtz 解的逼 近被獲得藉著滿足僅有的 Helmholtz 方程 的內邊界和外邊界條件。分片特解適合解 多介面與奇異的特徵值問題且達到高精度 節省計算量。 2

k

三、結果與討論 我們考慮特徵值方程：

⎩

⎨

⎧

Ω

∂

=

Γ

=

Φ

Ω

Φ

=

ΔΦ

−

on

0

in

i i i i

λ

(1) 這裡的

Ω

是多邊形有外邊界

Γ

。令特徵值 排序如下：

0

<

λ

₁

≤

λ

₂

≤

L

≤

λ

_i

≤

L

。特 徵函數

Φ

_i滿足正交性： ij j i j i

Φ

=

Φ

Φ

d

Ω

=

δ

Φ

_∫∫

Ω

)

,

(

， (2)

1

=

ij

δ

假如

i

=

j

；

δ

_ij

=

0

假如

i

≠

j

。令 解區域

Ω

被直線

Γ

₀分成兩個子區域

Ω

+與 −

Ω

則特徵函數必須滿足下列方程：

⎪

⎪

⎩

⎪⎪

⎨

⎧

Ω

∂

=

Γ

=

Φ

Γ

∂

Φ

∂

=

∂

Φ

∂

Φ

=

Φ

Ω

Ω

Φ

=

ΔΦ

−

− + − + − +

on

0

on

,

in

0 i i i i i i i i

n

n

U

λ

(3) 這裡的 是

n

Γ

₀單位法向量。定義一個輔助 Helmholtz 解

u

滿足下列方程：

⎪

⎪

⎩

⎪⎪

⎨

⎧

Γ

=

Γ

∂

∂

=

∂

∂

=

Ω

Ω

=

Δ

−

− + − + − +

on

on

,

in

0 2

g

u

n

u

n

u

u

u

u

k

u

_U

(4) 這裡的

k

>

0

且 是一個正函 數，也定義

)

(

2 / 1

_Γ

∈ H

g

2

k

與

λ

_i最小相對誤差為 2 2

min

k

k

_i i

λ

δ

=

−

。 Trefftz method: 首先定義

}

in

0

),

(

|

)

(

{

2 1 2 ± ±

Ω

=

+

Δ

Ω

∈

Ω

∈

=

v

k

v

H

v

L

v

H

∫

∫

∫

Γ − + Γ Γ − +

−

+

−

+

−

=

0 0 2 2 2 2

)

(

)

(

)

(

)

(

ds

v

v

ds

v

v

ds

g

v

v

I

n n

σ

∫

∫

∫

Γ − + − + Γ Γ − + − +

−

−

+

−

−

+

=

0 0

)

)(

(

)

)(

(

]

,

[

2

_u

_u

_v

_v

_ds

ds

v

v

u

u

uvds

v

u

n n n n

σ

2 2 2 2 0 0

|

|

|

|

|

|

]

,

[

|

|

Γ − + Γ − + Γ

+

−

+

−

=

=

n n B

v

v

v

v

v

v

v

v

σ

2 , 1 2 , 1

||

||

||

||

||

||

v

H

=

v

_Ω+

+

v

_Ω− 2 , 1 2 , 1

|

|

|

|

|

|

v

H

=

v

_Ω+

+

v

_Ω− 這裡的

H

1

(

Ω

±

)

是 Sobolev 空間且

σ

是正 的加權數。也定義有限維空間

S

_m,n

⊆

H

，

}

in

,

in

|

{

, − − − + + +

Ω

Ψ

=

=

Ω

Ψ

=

=

=

∑

∑

n i i i m i i i n m

d

v

v

c

v

v

v

S

這裡的 是方程式(4)的完備特解分 別在 且 與 是係數。當

}

{

Ψ

_i± ±

Ω

c

_i

d

_i

δ

>

0

，這 Trefftz 法是去發現

u

_m,n

∈

S

_m,n使得

)

(

min

)

(

, ,

I

v

u

I

n m S v n m

=

_∈ 。 (5) 這方程式(5)導出線性代數系統

Ax

=

b

，這 裡的

x

是全部的展開係數 與 組成的未 知向量。Trefftz 法也能被表示成弱解形式 且剛度矩陣 i

c

d

_i n m n m

v

gvds

v

S

u

_,

,

]

_,

[

=

∫

∀

∈

Γ

A

是非負正定與對稱被給在 n m T n m n m n m

u

x

Ax

u

_{, , , ,}

2

1

]

,

[

=

。 Helmholtzz 方程解為 1 , ,

c

u

u

mn

=

mn , 1 , ,

c

x

x

mn

=

mn ,

0

1

≠

c

。 Iterative Algorithms: Step 1. 選擇適當的

m,

n

與 3 個初始值 i i

k

≈

λ

，

i

=

0

,

1

,

2

。 Step 2. 由 Trefftz 法解

u

m,n並計算最小特 徵值

f

(

k

_i

)

=

λ

_min

(

A

(

k

_i

))

=

d

_min2 。 Step 3. 使用二次函數

P

₂

(

k

)

逼近

f

(k

)

通 過

(

k

_i

,

f

(

k

_i

))

，

i

=

n

,

n

−

1

,

n

−

2

，這裡的 皆不相同。計算新的 i

k

]

,

,

[

2

]

,

[

2

_{1 2} 1 1 1 − − − − +

=

+

−

n n n n n n n n

_f

_k

_k

_k

k

k

f

k

k

k

2

≥

n

，這裡的 1 1 1

)

(

)

(

]

,

[

− − −

=

₋

−

n n n n n n

_k

_k

k

f

k

f

k

k

f

2 2 1 1 2 1

]

,

[

]

,

[

]

,

,

[

− − − − − −

=

₋

−

n n n n n n n n n

_k

_k

k

k

f

k

k

f

k

k

k

f

Step 4. 假如

f

(k

)

足夠小則 是一個好 的逼近到 2

k

i

λ

且

u

m,n逼近到 ；否則重複 Step 2-4。假如無法得到好的逼近則回到 Step 1 增加 重新來過。 i

Φ

n

m,

Error Analysis: 我們使用輔助函數

w

滿足下列方程:

⎪

⎪

⎩

⎪⎪

⎨

⎧

Γ

=

Γ

=

∂

∂

−

∂

∂

=

−

Ω

Ω

=

Δ

− + − + − +

on

on

,

in

0

0 2 1

g

w

n

w

n

w

w

w

w

ε

ε

U

Theorem 1: Let be the piecewise

particular solution of the Helmholtz equation (4) and the auxiliary function satisfy

u

w

Ω Ω

≤

0, , 0

|

|

2

1

|

|

w

u

and the inverse

properties and

. Then there exists

H w n

K

w

w

|

||

||

|

_0,Γ

≤

H w n

K

w

w

|

||

||

|

0 , 0Γ

≤

+ i

λ

such that Ω −

+

≤

−

, 0 1 2 2

|

|

|

|

)

(

|

|

u

u

K

C

k

k

_B w i

_σ

λ

這裡的

K

_w是與

w

有關的常數。

Theorem 2: Let the conditions in Theorem 1 hold and

|

λ

_i

−

λ

_j

|

≥

β

>

0

,

i

≠

j

and

Ω Ω

<

₂ 0, , 0

}

|

|

4

2

1

min{

|

|

u

k

β

,

w

. Then

there exists a real constant

a

_i

≠

0

such that

B w i i i

C

K

u

a

u

|

(

)

|

|

|

−

Φ

_0,Ω

≤

+

σ

−1

β

λ

.

Theorem 3: Let the conditions in Theorem 2 hold. Then there exists a real constant

such that

0

≠

i

a

B w i H i i

C

K

u

a

u

||

(

)

|

|

||

1 2 / 3 −

+

≤

Φ

−

σ

β

λ

四、計劃成果自評 在參考文獻 II 與 III 中，我們列出自 2005 年已發表或被錄取的期刊論文計有 17 篇[7-23]，以及會議論文 6 篇[24-29]。其中 與 Trefftz 法及邊界方法相關期刊論文有 8 篇[9，11，13，16，17，19，21，22]，而 與 Trefftz 法有關的會議論文有 6 篇[24-29]。 五、參考文獻 I. References

[1] S. Bergman, Integral Operators in the Theory of Linear Partial Differential

Equations, Springer-Verlag, Berlin, New York, 1969.

[2] S.C. Eisenstat, On the note of

convergence of the Bergman-Vekua methods for the numerical solution of elliptic boundary-value problems, SIAM J. Numer. Anal., vol.11, pp. 654-680, 1974.

[3] L. Fox, P. Henrici and C. Moler,

Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal., vol. 4, pp. 89-102, 1967.

[4] R. Mathon and P. Sermer, Numerical

solution of the Helmholtz equation, Congressus Numerantium, vol. 34, pp. 313-330, 1982.

[5] I.N. Vekua, New Method for Solving

Ellitic Equations, North-Holland, Amsterdam, New York, 1967.

[6] Z.C. Li, R. Mathon and P. Sermer,

Boundary methods for solving elliptic problems with singularities and interfaces, SIAM J. Numer. Anal., vol. 24, pp. 487-498, 1987.

II. Refereed Journal Papers

[7] H.Y.Hu and Z.C.Li (2005),

Verification of reduced convergence rates,Computing, vol. 74, pp.67-73, 2005. [SCI]

[8] Z.C.Li, T.T.Lu, H.Y.Hu and

A.H.D.Cheng (2005), Particular solutions of Laplace's equations on polygons and new models involving mild singularities, Engineering Analysis with Boundary Elements, vol. 29, pp. 59-75, 2005. [SCI]

[9] T.T.Lu and Z.C.Li (2005), The

cracked-beam problem by boundary approximation method Applied Mathematics Letter, vol. 18, pp. 11-16, 2005. [SCI]

[10] Z.C.Li (2005), Algorithms for curve

image under geometric transformations, Inter. J. Information, vol.8, pp.845-862.

[11] Z.C.Li, C.S.Huang and R.C.D.Chen (2005), Interior boundary conditions in the Schwarz alternating method for the Trefftz method, Engineering Analysis with Boundary Elements, vol. 29, pp. 477-493, 2005. [SCI]

[12] H.Y.Hu, Z.C.Li and A.H.-D.Cheng

(2005), Radial basis collocation methods for elliptic boundary value problems, Inter. J. Computers & Mathematics with Application, vol. 50, pp.289-320, 2005.

[13] Z.C.Li, Y.L.Chan, G.G.Georgiou and

X.Xenophotos (2006), Special Boundary Approximation methods for Laplace equation problems with boundary singularities (applications to Motz problem), Inter. J. Computers & Mathematics with Application, vol. 51, pp. 115, 142, 2006. [SCI]

[14] H.T.Huang and Z.C.Li and A.Zhou

(2006), New error estimates of Biquadratic Lagrange Elements for Poisson's equation, Applied Numerical Mathematics, vol. 56, pp. 712-744, 2006. [SCI]

[15] H.Y.Hu and Z.C.Li (2006), Collocation

Methods for Poisson's equation, vol. 195, pp. 4139-4160, 2006. [SCI]

[16] H.T.Huang and Z.C.Li (2006), Effective condition number and superconvergence of the Trefftz method coupled with high order FEM for singularity problems, vol. 30, pp. 270-283, 2006. [SCI]

[17] Z.C.Li, T.T.Lu, H.S.Tsai, and

A.H.-D.Cheng (2006), The Trefftz method for eigenvalue problems, Comput. Method Appl. Mech. Eng., vol. 39, pp. 292-308, 2006. [SCI]

[18] H.Y.Hu, and Z.C.Li, (2006),

Combinations of collocation and finite element methods for Poisson's equation, Inter. J. Computers & Mathematics with Application, vol. 51, pp. 1831-1853.

[19] Z.C.Li, T.T. Lu, H.T.Huang and

A.H.-D.Cheng (2006), Trefftz, collocation and other boundary methods -- A comparison, accepted by Numer. Methods for PDEs.[JCR]

[20] Z.C.Li, C.S.Chien and H.T.Huang

(2007), Effective Condition number for finite difference method, J. Comput. and

Applied, vol. 198, pp. 208-235, 2007. [SCI]

[21] Z.C.Li (2007), Error analysis of the

Trefftz method for solving Laplace's eigenvalue problems, accepted by J. Computational and Applied Mathematics.

[22] Z.C.Li, H.T.Huang, J.Hunag and L.Ling

(2007), Stability analysis for the penalty plus hybrid and the direct Trefftz method for singularity problems, accepted by Engineering Analysis with Boundary Elements.[SCI]

[23] C.S.Huang, C.S.Wang, C.S.Chen and

Z.C.Li (2007), A radial collocation method for Hamilton-Jacobi-Bellman equation, accepted by Automatica. [SCI]

III. Conference Papers

[24] Z.C.Li, T.T.Lu and H.S.Tsai, The

Trefftz method for solving Laplace eigenvalue problems,,Inter. Conf. on Bounary Element Techniques VI, July, 2005, Montreal, Canada.

[25] Z.C.Li and H.T.Huang, High

superconvergence of combinations of Trefftz method and FEMs for Poisson’s equation with singularities, Inter. Conf. on Bounary Element Techniques VI, July, 2005, Montreal, Canada.

[26] H.T.Huang and Z.C.Li, Coupling

techniques of Trefftz method, Inter. Conf. on Bounary Element Techniques VI, July, 2005, Montreal, Canada.

[27] Z.C.Li (2006), Effective Condition

Number for Trefftz Methods, 7th World congress on computational mechanics, July, 2006, LA, USA.

[28] H.T.Huang and Z.C.Li, New stability

analysis for Trefftz methods coupled with high order FEM for singularity problems, 7th World congress on computational mechanics, July, 2006, LA, USA.

[29] T.Liu, Z.C.Li, H.T.Huang and

A.H.-D.Cheng, Trefftz, Collocation and other boundary methods -- A comparison, 7th World congress on computational mechanics, July, 2006, LA, USA