EFFICIENT TREFFTZ COLLOCATION ALGORITHMS

EFFICIENT TREFFTZ COLLOCATION ALGORITHMS FOR ELLIPTIC PROBLEMS IN CIRCULAR DOMAINS A. KARAGEORGHIS

Abstract. We consider the numerical solution of certain elliptic boundary value problems in disks and annuli using the Trefftz collocation method. In particular we examine boundary value problems for the Laplace, Helmholtz, modified Helmholtz and biharmonic equations in such domains. It is shown that this approach leads to systems in which the matrices possess specific structures. By exploiting these structures we propose efficient algorithms for the solution of the systems. The proposed algorithms are applied to standard test problems.

1. Introduction Trefftz methods [30] have been used extensively for the solution of elliptic boundary problems; see, for example [21, 22, 31, 33]. Surveys on Trefftz and related methods may be found in [17, 20, 23]. The general idea in these methods is to approximate the solution of a boundary value problem by a linear combination of basis functions which satisfy the differential equation of the problem in question. The coefficients in the expansion are determined by satisfying the boundary conditions in some sense. Usually, this is done using collocation and the basis functions are taken to be either fundamental solutions of the operator in the differential equation with singularities placed outside the domain of the problem or the so-called T-complete functions [15, 16]. The former choice leads to the method of fundamental solutions (MFS) [10] and has become extremely popular in recent years in particular for the numerical solution of problems in complex geometries and three dimensions as well as inverse problems [18]. The latter approach is known as the Trefftz collocation method (TCM) [23] and has also been applied extensively to inverse problems [2, 5, 6, 7, 8, 11, 12, 19, 26, 32]. Both methods belong to the class of boundary meshless methods since only the boundary of the domain under consideration needs to be discretized and no mesh is generated. This makes them easy to implement, a very attractive feature especially when dealing with problems involving complex or unknown boundaries (such as inverse problems). One disadvantage of the MFS over the TCM is the fact that the optimal location of the singularities is unknown. On the other hand the TCM leads to poorly conditioned systems. Efforts to address this issue include the normalization of the coefficients in the expansion [24, 25] and regularization techniques [34]. Some equivalence results between the MFS and the TCM can be found in [3, 4]. In this work we examine the application of the TCM to elliptic problems in disks and annuli. In the case of Laplace, Helmholtz and modified Helmholtz equations the approach is very similar and in each case the structure of the resulting linear systems is exploited to yield efficient algorithms which make use of fast Fourier transforms (FFTs) for the solution of problems in such domains. The same ideas are subsequently used for the efficient solution of biharmonic boundary value problems in disks and annuli. In Section 2 we describe the TCM when applied to harmonic, Helmholtz and modified Helmholtz problems in disks and annuli and provide the resulting systems of linear equations. In Section 3 we present two algorithms, one for the solution of systems of linear equations in which the matrices possess a specific structure and the other for the efficient matrix-vector multiplication involving these matrices. Both of these operations are used repeatedly in the proposed algorithms for the efficient solution of the problems of Section 2, as is described in Section 4. Date: November 15, 2012. 2000 Mathematics Subject Classification. Primary 65N35; Secondary 65N99. Key words and phrases. Trefftz method, collocation, Fast Fourier transforms.

1

2

A. KARAGEORGHIS

The application of the proposed algorithms to some standard test problems is considered in Section 5 while their extension to the efficient solution of biharmonic problems as well as some numerical examples are presented in Section 6. Finally some concluding remarks are given in Section 7.

2. Trefftz collocation method 2.1. Problems in disks. We consider the boundary value problem ∆u + κu = 0

in Ω,

(2.1a)

∂Ω,

(2.1b)

subject to the Dirichlet boundary condition on

u = f or the Neumann boundary condition

∂u = f on ∂Ω, (2.1c) ∂n where the domain Ω is a disk of radius R centred at the origin. When κ = 0 equation (2.1a) is the Laplace equation, when κ > 0 equation (2.1a) is the Helmholtz equation while when κ < 0 equation (2.1a) is the modified Helmholtz equation. Note than in the case of the Laplace equation we cannot have only the Neumann boundary condition (2.1c). In the TCM for simply connected domains, we seek an approximation to the solution of (2.1a) as a linear combination of T-complete functions in the form [23] uN (α0 , α, β; x) = α0 φ0 (r) +

N ∑

N ∑

αm φm (r) cos mϑ +

m=1

βm φm (r) sin mϑ,

x = r(cos ϑ, sin ϑ) ∈ Ω,

(2.2)

m=1

where the T-complete system is given by φm (r) = rm ,

m = 0, . . . , N,

in the case of the Laplace equation,

φm (r) = Jm (κr), m = 0, . . . , N, in the case of the Helmholtz equation, φm (r) = Im (κr), m = 0, . . . , N, in the case of the modified Helmholtz equation,

(2.3)

where Jm (r) is the Bessel function of the first kind and Im (r) is the modified Bessel function of the first kind. In (2.2) there are 2N + 1 unknowns, namely the coefficients α0 , α = [α1 , . . . , αN ]T and β = [β1 , . . . , βN ]T . Since there are 2N + 1 unknowns, it is natural to choose the boundary collocation points as xn = R (cos ϑn , sin ϑn ) ,

n = 1, . . . , 2N + 1,

(2.4)

2π(n − 1) respectively, where ϑn = , n = 1, . . . , 2N + 1. 2N + 1 Collocation of the boundary condition (2.1b) yields uN (α0 , α, β; xn ) = f (xn ),

n = 1, . . . , 2N + 1,

while collocation of the boundary condition (2.1c) yields ∂uN (α0 , α, β; xn ) = f (xn ), n = 1, . . . , 2N + 1. ∂n In either case we obtain a (2N + 1) × (2N + 1) system of the general form Ax = f , where



µ A= µ µ

λT A C

 0T B , D



 α0 x =  α , β

(2.5)

(2.6)

(2.7) 

 f0 f =  f1  . f2

(2.8)

EFFICIENT TREFFTZ COLLOCATION ALGORITHMS

3

In (2.8) µ is a non-zero constant, the N × 1 vector µ is defined by µ = µhN ,where the N × 1 vector hN = [1, 1, . . . , 1]T , the N × 1 vector λ is defined by λ = [λ1 , λ2 , . . . , λN ]T and the N × N matrices A, B, C and D are defined by ( ) ( ) 2πmn 2πmn N,N N,N A = [amn ]m=1,n=1 , amn = λn cos , B = [bmn ]m=1,n=1 , bmn = λn sin , 2N + 1 2N + 1 ( ) ( ) 2π(N + m)n 2π(N + m)n N,N N,N C = [cmn ]m=1,n=1 , cmn = λn cos , D = [dmn ]m=1,n=1 , dmn = λn sin . (2.9) 2N + 1 2N + 1 The the N × 1 vectors α, β, f 1 , f 2 are defined by α = [α1 , α2 , . . . , αN ]T , β = [β1 , β2 , . . . , βN ]T and f ℓ = [fℓ1 , fℓ2 , . . . , fℓN ]T , ℓ = 1, 2. More specifically, µ = φ0 (R),

λn = φn (R) in the case of the Dirichlet boundary condition (2.1b),

µ = φ′0 (R),

and

λn = φ′n (R) in the case of the Neumann boundary condition (2.1c).

(2.10)

2.2. Problems in annuli. We now consider the boundary value problem ∆u + κu = 0

in Ω,

(2.11a)

subject to the boundary conditions on ∂Ω1 ,

u = f1

(2.11b)

or ∂u = f1 ∂n

on ∂Ω1 ,

(2.11c)

and on ∂Ω2 ,

u = f2

(2.11d)

or ∂u = f2 on ∂Ω2 , (2.11e) ∂n where the domain Ω is an annulus of radii R1 and R2 (R2 > R1 ), corresponding to the boundaries ∂Ω1 and ∂Ω2 , respectively. In the TCM for doubly connected domains, we seek an approximation to the solution (2.11a) as a linear combination of T-complete functions in the form [11, 23, 25] uN (α0 , α, β, γ0 , γ, δ; x) = α0 φ0 (r) +

N ∑


m=1

+γ0 χ0 (r) +

N ∑

γm χm (r) cos mϑ +

m=1

N ∑

βm φm (r) sin mϑ

m=1 N ∑

δm χm (r) sin mϑ,

x = (x, y) ∈ Ω,

(2.12)

m=1

where the functions φm (r) are defined as in (2.10) and χm (r) = r−m , m = 0, . . . , N, in the case of the Laplace equation, χm (r) = Ym (κr), m = 0, . . . , N, in the case of the Helmholtz equation, χm (r) = Km (κr), m = 0, . . . , N, in the case of the modified Helmholtz equation,

(2.13)

where Ym (r) is the Bessel function of the second kind and Km (r) is the modified Bessel function of the second kind. In (2.12) there are 4N + 2 unknowns, namely the coefficients α0 , α = [α1 , . . . , αN ]T , β = [β1 , . . . , βN ]T , γ0 , γ = [γ1 , . . . , γN ]T and δ = [δ1 , . . . , δN ]T .

4

A. KARAGEORGHIS

Since there are 4N + 2 unknowns, it is natural to choose the boundary collocation points as xn = R1 (cos ϑn , sin ϑn ) , respectively, where ϑn =

and

x2N +1+n = R2 (cos ϑn , sin ϑn ) ,

n = 1, . . . , 2N + 1,

(2.14)

2π(n − 1) , n = 1, . . . , 2N + 1. 2N + 1

We shall consider the following three cases of collocating boundary conditions for n = 1, . . . , 2N + 1: Dirichlet case: uN (α0 , α, β, γ0 , γ; xn ) = f1 (xn )

and

uN (α0 , α, β, γ0 , γ; x2N +1+n ) = f2 (x2N +1+n ), ∂uN Mixed case: uN (α0 , α, β, γ0 , γ; xn ) = f1 (xn ) and (α0 , α, β, γ0 , γ; x2N +1+n ) = f2 (x2N +1+n ), ∂n ∂uN ∂uN Neumann case: (α0 , α, β, γ0 , γ; xn ) = f1 (xn ) and (α0 , α, β, γ0 , γ; x2N +1+n ) = f2 (x2N +1+n ). (2.15) ∂n ∂n In each of the three cases we obtain a (4N + 2) × (4N + 2) system of the form ( )( ) ( ) A11 A12 x1 f1 = , A21 A22 x2 f2

(2.16)

where the (2N + 1) × (2N + 1) matrices A11 , A12 , A21 , A22 have the form of matrix A in (2.7) with µ and λn defined by Dirichlet case: µ11 = φ0 (R1 ), λ11 n = φn (R1 ), 21

µ Mixed case:

= φ0 (R2 ),

µ11 = φ0 (R1 ), µ21 = φ′0 (R2 ),

Neumann case:

µ11 = φ′0 (R1 ), µ21 = φ′0 (R2 ),

λ21 n = φn (R2 ), λ11 n = φn (R1 ), ′ λ21 n = φn (R2 ), ′ λ11 n = φn (R1 ), ′ λ21 n = φn (R2 ),

µ12 = χ0 (R1 ), λ12 n = χn (R1 ), µ22 = χ0 (R2 ), λ22 n = χn (R2 ), µ12 = χ0 (R1 ), λ12 n = χn (R1 ), ′ µ22 = χ′0 (R2 ), λ22 n = χn (R2 ), ′ µ12 = χ′0 (R1 ), λ12 n = χn (R1 ), ′ µ22 = χ′0 (R2 ), λ22 n = χn (R2 ),

(2.17)

respectively. Also, the (2N + 1) × 1 vectors xℓ , fℓ , ℓ = 1, 2, are defined as xT1 = [α0 , αT , β T ],

xT2 = [γ0 , γ T , δ T ],

fℓT = [fℓ0 , f Tℓ1 , f Tℓ2 ], ℓ = 1, 2,

where α, β are defined as in (2.8), γ = [γ1 , γ2 , . . . , γN ]T , δ = [δ1 , δ2 , . . . , δN ]T , and f ℓk = [fℓk 1 , fℓk 2 , . . . , fℓk N ]T , ℓ, k = 1, 2, with fℓ0 = fℓ (x(ℓ−1)(2N +1)+1 ), fℓ1n = fℓ (x(ℓ−1)(2N +1)+1+n ), fℓ2n = fℓ (x(ℓ−1)(2N +1)+N +1+n ), ℓ = 1, 2, n = 1, . . . , N .

3. Efficient algorithms The algorithms proposed in this study are based on two specific operations involving certain matrices, namely the efficient solution of systems of linear equations and the efficient matrix-vector multiplication. 3.1. Efficient solution of systems. We first consider the efficient solution of (2N + 1) × (2N + 1) systems of the form (2.7). We can re-write system (2.7) as 

1  hN hN

hTN Aˆ Cˆ

    0T α ˆ0 f0 ˆ  α ˆ  =  f1  , B ˆ ˆ f2 β D

(3.1)


5

ˆ = λ · β and the N × N matrices A, ˆ B, ˆ Cˆ and D ˆ are defined by ˆ = λ · α, β where α ˆ 0 = µα0 , α ( ) ( ) [ ] N,N 2πmn 2πmn N,N ˆ ˆ ˆ ˆ A = [ˆ amn ]m=1,n=1 , a ˆmn = cos , B = bmn , bmn = sin , 2N + 1 2N + 1 m=1,n=1 ( ( ) ) [ ]N,N 2π(N + m)n 2π(N + m)n N,N ˆ = dˆmn Cˆ = [ˆ cmn ]m=1,n=1 , cˆmn = cos , D , dˆmn = sin . (3.2) 2N + 1 2N + 1 m=1,n=1 We observe that cˆmn = a ˆN −m+1,n Hence we may rearrange system (3.1) as  1  hN hN

and hTN Aˆ Aˆ

dˆmn = −ˆbN −m+1,n  0T ˆ  B ˆ −B

for m, n = 1, . . . N.

   α ˆ0 g0 ˆ  =  g1  , α ˆ g2 β

(3.3)

(3.4)

where g0 = f0 , g 1 = f 1 , g 2 = [f2N , f2N −1 , . . . , f21 ]T . System (3.4) can be written as ˆ = g0 , α ˆ 0 + hTN α

(3.5a)

ˆ =g ˆβ ˆ +B α ˆ 0 hN + Aˆα 1

(3.5b)

ˆ =g . ˆβ ˆ −B α ˆ 0 hN + Aˆα 2

(3.5c)

and Subtracting (3.5c) from (3.5b) we obtain ˆ = 1 (g − g ) . ˆβ B 2 2 1 ˆ Since the matrix B has the property (see Appendix) ˆB ˆ = 2N + 1 IN B 4 where IN is the N × N identity matrix, system (3.6) yields 2 ˆ= ˆ (g 1 − g 2 ) . B β 2N + 1 Adding (3.5c) and (3.5b) we obtain ˆ = 1 (g 1 + g 2 ) α ˆ 0 hN + Aα 2 ˆ from (3.5a) we have and since α ˆ 0 = g0 − hTN α 1 ˆ − hTN αh ˆ N = (g 1 + g 2 ) − α ˆ 0 hN , Aˆα 2 or ( ) 1 ˆ = (g 1 + g 2 ) − α ˆ 0 hN , Aˆ − H α 2 where all the entries of the N × N matrix H are equal to one, i.e. H = hN hTN . Since the matrix Aˆ − H has the property (see Appendix) ( )( ) 1 N 1 Aˆ − H Aˆ − H = (N + )H + ( + )IN = G 2 2 4 system (3.11) is equivalent to the system ( ) ˆ = Aˆ − H g ˆ = a, Gα (1 ) ˆ = 2 (g 1 + g 2 ) − α where g ˆ 0 hN . If cN = (2N + 1)/4, then G = cN IN + 2cN H = cN IN + 2cN hN hTN .

(3.6)

(3.7)

(3.8)

(3.9)

(3.10) (3.11)

(3.12)

(3.13)

(3.14)

6

A. KARAGEORGHIS

Using the Sherman-Morrison-Woodbury formula [13, page 50] we obtain the inverse of the matrix G G−1 = c−1 N IN − and hence

2c−1 N H, 2N + 1

( ) 2c−1 −1 N ˆ = cN IN − H a. α 2N + 1

(3.15)

(3.16)

Alternatively, we observe that the matrix G is circulant [9] and thus system (3.13) may be solved efficiently using ( )N a matrix decomposition algorithm [29]. If we define the matrix U ∈ CN ×N by U = √1N e−2πi(m−1)(n−1)/N m,n=1 , i2 = −1, we premultiply system (3.13) by U to obtain ˆ = U a or Dα ˜ =a ˜, U GU ∗ U α ˜ = U a, α ˜ = Uα ˆ and the matrix D = U GU ∗ is diagonal. The elements of α ˜ = [α where a ˜1, α ˜2, . . . , α ˜ N ] can be ˆ can be recovered from α ˆ = U ∗ α. ˜ Note that the easily calculated from α ñ = a ñ /Dnn , n = 1, . . . , N and then α ˜ are carried out efficiently using FFTs (for details, see [29]). The ˆ = U a, D = U GU ∗ and α ˆ = U ∗α operations a cost of solving system (3.13) is thus O(N log N ). Finally, the solution of system (2.7) is recovered from α0 = α ˆ 0 /µ, αn = α ˆ n /λn , βn = βˆn /λn , n = 1, . . . , N . The efficient solution algorithm for solving (3.4) can be summarized as follows: Algorithm 1. Step 1. Obtain g0 , g 1 , g 2 from f0 , f 1 , f 2 . ˆ from (3.8). Step 2. Compute β ˆ by solving system (3.13). Step 3. Compute α Step 4. Evaluate α ˆ 0 from (3.5a). ˆ ˆ β. Step 5. Recover α0 , α, β from α ˆ 0 , α, Cost. This algorithm requires O(N log N ) operations. In particular, in Step 2 the most expensive part is the ˆ (g 1 − g 2 ). However, because of the structure of B ˆ this can be carried out using evaluation of the right hand side B ( ) ˆ may FFTs at a cost of O(N log N ) operations. In Step 3 the evaluation of the right hand side a = Aˆ − H g ˆg and H g ˆg because of the structure of Aˆ can be carried ˆ . The evaluation of Aˆ be split up in the evaluation of Aˆ ˆ is trivial because of the structure of H out using FFTs at a cost of O(N log N ) operations. The evaluation of H g ˆ are equal to the sum of the elements of the vector g ˆ (hence which means that all the elements of the vector H g calculated only once). The solution of system (3.13) can be carried out using the Sherman-Morrison-Woodbury formula which yields the expression (3.16) for the solution. Clearly, because of the structure of the inverse of G ˆ from (3.16) (once the right hand side a has been evaluated) is O(N ). Alternatively, the cost of calculating α solution of system (3.13) can be carried out via a matrix decomposition algorithm, as explained earlier, at a cost of O(N log N ). All the FFT operations are carried out using the Matlab [27] command fft. 3.2. Efficient matrix-vector multiplication. We now consider the efficient calculation of the matrix-vector form p = Ax,

(3.17)

where the (2N +1)×(2N +1) matrix A and the (2N +1)×1 vector x are defined as in (2.8), and p = [p0 , p1 , . . . , p2N ]T .


We can re-write (3.17) as



µ hTN  p= µ Aˆ µ Cˆ

 0T ˆ  B ˆ D

 α0 ˆ , α ˆ β

7

(3.18)

ˆ and the matrices A, ˆ B, ˆ Cˆ and D ˆ are defined as in (3.1). Using (3.3) we may rearrange system (3.18) as ˆ β where α,    µ hTN 0T α0 ˆ  α ˆ , ˆ =  µ Aˆ p (3.19) B ˆ ˆ ˆ β µ A −B ˆ 1 = [ˆ ˆ 2 = [ˆ where pˆ0 = p0 , pˆm = pm , pˆN +m = p2N −m+1 , m = 1, . . . , N . With p p1 , pˆ2 , . . . , pˆN ]T , p pN +1 , pˆN +2 , . . . , pˆ2N ]T , (3.19) can be decomposed as ˆ pˆ0 = µα0 + hTN α, (3.20a) ˆ ˆ β, ˆ = α0 µ + Aˆα ˆ +B p (3.20b) 1

ˆ ˆ β. ˆ 2 = α0 µ + Aˆα ˆ −B p ˆ has been obtained, p is recovered from p0 = pˆ0 , pm = pˆm , pN +m = pˆ2N −m+1 , m = 1, . . . , N . Once p

(3.20c)

Algorithm 2. Step 1. Transform (3.17) into the form (3.18). Step 2. Rearrange (3.18) into the form (3.19). ˆ and hence the vector p ˆβ ˆ and B ˆ. Step 3. Compute Aˆα ˆ to obtain the vector p. Step 4. Rearrange the vector p Cost. This algorithm requires O(N log N ) operations. The most expensive part of the algorithm occurs in Step ˆ However, because of the structures of B ˆ β. ˆ and Aˆ these each can be carried ˆ and B 3 with the multiplications Aˆα out using FFTs at a cost of O(N log N ) operations. 4. Efficient solution of problems in disks and annuli We next consider the efficient solution of problems in disks and annuli which led to systems (2.7) and (2.16), respectively. 4.1. Problems in disks. We simply use Algorithm 1 described in Section 3.1 to solve system (2.7) at a cost of O(N log N ). 4.2. Problems in annuli. In each of the three cases (2.17) system (2.16) can be written as     α ˆ0 f1 0  f 11  ˆ  ( )  α    ˆ ˆ ˆ     A11 A12  β  =  f 12  ,     Aˆ21 Aˆ22  γˆ0   f2 0   γ   ˆ f 21  f 22 δˆ

(4.1)

where the (2N + 1) × (2N + 1) matrices Aˆ11 , Aˆ12 , Aˆ21 , Aˆ22 have the form of matrix A in (2.7) with µ and λn defined as λ21 λ22 µ21 ˆ 21 µ22 ˆ 22 n n 12 21 22 ˆ 11 = 1, µ ˆ 12 = 1, µ , λ = , λ = µ ˆ11 = 1, λ ˆ = 1, λ ˆ = , µ ˆ = , n n µ11 n λ11 µ12 n λ12 n n ˆ = λ11 · β, γˆ0 = µ12 γ0 , γ ˆ = λ11 · α, β ˆ = λ12 · γ and δˆ = λ12 · δ. respectively. Also, α ˆ 0 = µ11 α0 , α

8

A. KARAGEORGHIS

Since Aˆ11 , = Aˆ12 system (4.1) may be written as (

Aˆ11 Aˆ

21

0 A˜

22



)      

α ˜0 ˜ α ˜ β γˆ0 ˆ γ δˆ





       =       

f1 0 f 11 f 12 f2 0 f 21 f 22

    ,   

(4.2)

µ22 µ21 ˜ 22 λ22 λ21 n n where the matrix A˜22 is of the form of matrix A in (2.7) with µ and λn defined as µ ˜22 = 12 − 11 , λ n = 12 − 11 , µ µ λn λn ˜ =β ˆ + δ. ˆ ˜ =α ˆ +γ ˆ, β respectively. In addition, α ˜0 = α ˆ 0 + γˆ0 , α System (4.2) may be decomposed as 

and

   α ˜0 f1 0 ˜  =  f 11  , Aˆ11  α ˜ f 12 β

(4.3)

     γˆ0 α ˜0 f2 0 ˆ  =  f 21  . ˜  + A˜22  γ Aˆ21  α ˜ f 22 β δˆ

(4.4)



System (4.3) has the structure of system (2.7) and hence can be solved efficiently using Algorithm 1 at a cost of ˜ Then, from (4.4) we obtain ˜ and β. O(N log N ) to yield α ˜0, α       γˆ0 α ˜0 f2 0 ˆ  =  f 21  − Aˆ21  α ˜ . A˜22  γ (4.5) ˜ ˆ f β δ 22 System (4.5) also has the structure system (2.7) and can thus be solved using Algorithm 1 at a cost of O(N log N ) ˜ In addition, since the matrix Aˆ21 has the form of matrix A in (3.17) the multipli˜ and δ. operations to yield γ˜0 , γ cation in the second term of the right hand side can also be carried out at a cost of O(N log N ) using Algorithm 2. ˜ we obtain α ˆ = β ˜ − δ. ˆ Finally, we recover the solution from ˜ and β ˆ = α ˜ −γ ˆ and β From α ˜0, α ˆ0 = α ˜ 0 − γˆ0 , α 11 11 11 12 12 ˆ α0 = α ˆ 0 /µ , αn = α ˆ n /λn , βn = βn /λn ,γ0 = γˆ0 /µ , γn = γˆn /λn and δn = δˆn /λ12 n , n = 1, . . . , N .

5. Numerical examples In the numerical examples involving the Helmholtz and modified Helmholtz equations, the Bessel functions Jk , Yk , Ik and Kk , k = 0, . . . N , are calculated using the Matlab [27] functions besselj, bessely, besseli and besselk, respectively. In addition, the derivatives of these functions are calculated using the properties [1] J0′ (r) = −J1 (r), Y0′ (r) = −Y1 (r), I0′ (r) = I1 (r), K0′ (r) = −K1 (r), 1 1 ′ Jm (r) = (Jm−1 (r) − Jm+1 (r)) , Ym′ (r) = (Ym−1 (r) − Ym+1 (r)) , 2 2 1 1 ′ ′ Im (r) = (Im−1 (r) + Im+1 (r)) , Km (r) = − (Km−1 (r) + Km+1 (r)) , m = 1, . . . , N. 2 2

(5.1)


9

5.1. Example 1. We first consider the Dirichlet problem (2.1a)-(2.1b) for the Laplace equation in a disk of radius R = 2 with the boundary condition corresponding to the exact solution u(x, y) = (sin x + cos x)ey .

(5.2)

Since the approximation is harmonic it is sufficient to calculate the error on the boundary. We calculated the maximum absolute error EN in the approximation uN for various values of N on a set of 100 uniformly distributed points on ∂Ω (different than the collocation points). In Figure 1 we present the maximum absolute error EN versus N. Error vs N

0

10

−2

10

−4

10

−6

EN

10

−8

10

−10

10

−12

10

−14

10

4

6

8

10

12 N

14

16

18

20

Figure 1. Example 1: Maximum absolute error versus N .

5.2. Example 2. We consider the Dirichlet problem (2.11a), (2.11b), (2.11d) for the Laplace equation in an annulus of radii R1 = 1/2, R2 = 2 with the boundary conditions corresponding to the exact solution (5.2). We calculated the maximum absolute errors EℓN , ℓ = 1, 2 in the approximation uN for various values of N on a set of 100 uniformly distributed points on ∂Ωℓ , ℓ = 1, 2, respectively (different than the collocation points). In Figure 2 we present the maximum absolute errors E1N , E2N versus N . 5.3. Example 3. We consider the mixed problem (2.11a), (2.11b), (2.11e) for the Laplace equation in an annulus of radii R1 = 1/2, R2 = 2 with the boundary conditions corresponding to the exact solution (5.2). We calculated the maximum absolute errors EℓN , ℓ = 1, 2 as in Example 2. In Figure 3 we present the maximum absolute errors E1N , E2N versus N . 5.4. Example 4. We consider the Neumann problem (2.1a), (2.1c) for the Helmholtz equation with κ = 1 in a disk of radius R = 2 with the boundary condition corresponding to the exact solution ( √ √ ) u(x, y) = sin( 2x) + cos( 2x) cosh y. (5.3)

10

A. KARAGEORGHIS

Error vs N

0

10

E1

N

E2

N

−5

Error

10

−10

10

−15

10

4

6

8

10

12 N

14

16

18

20

Figure 2. Example 2: Maximum absolute errors versus N . Error vs N

0

10

E1

N

E2

N

−5

Error

10

−10

10

−15

10

4

6

8

10

12 N

14

16

18

20

Figure 3. Example 3: Maximum absolute errors versus N . ¯ We therefore calculated the maximum absolute Now the error is no longer harmonic and needs to be evaluated in Ω. ¯ error EN in the approximation uN for various values of N on a set of 25 × 51 uniformly distributed points in Ω. In Figure 4 we present the maximum absolute error EN versus N .


11

Error vs N

2

10

0

10

−2

10

−4

Error

10

−6

10

−8

10

−10

10

−12

10

4

6

8

10

12 N

14

16

18

20


5.5. Example 5. We consider the mixed problem (2.11a), (2.11b), (2.11e) for the Helmholtz equation with κ = 1 in an annulus of radii R1 = 1/2, R2 = 2 with the boundary conditions corresponding to the exact solution (5.3). We calculated the maximum absolute error EN in the approximation uN for various values of N on a set of 25 × 51 ¯ In Figure 5 we present the maximum absolute error EN versus N . uniformly distributed points in Ω. 5.6. Example 6. We finally consider the Neumann problem (2.11a), (2.11c), (2.11e) for the modified Helmholtz equation with κ = −1 in an annulus of radii R1 = 1/2, R2 = 2 with the boundary conditions corresponding to the exact solution √ u(x, y) = (sin x + cos x) cosh( 2y). (5.4) We calculated the maximum absolute error EN as in Example 5. In Figure 6 we present the maximum absolute error EN versus N .

6. Extension to the biharmonic equation 6.1. Problems in disks. We consider the boundary value problem ∆2 u = 0

in Ω,

(6.1a)

subject to the boundary conditions u = f1 where the domain Ω is a disk of radius R.

and

∂u = f2 ∂n

on ∂Ω,

(6.1b)

12

A. KARAGEORGHIS

Error vs N

2

10

0

10

−2

10

−4

Error

10

−6

10

−8

10

−10

10

−12

10

4

6

8

10

12 N

14

16

18

20

Figure 5. Example 5: Maximum absolute error versus N . Error vs N

2

10

0

10

−2

10

−4

Error

10

−6

10

−8

10

−10

10

−12

10

4

6

8

10

12 N

14

16

18

20

Figure 6. Example 6: Maximum absolute error versus N . In the biharmonic TCM for simply connected domains, we seek an approximation to the solution of (6.1a) as a linear combination of T-complete functions in the form [21, 23] uN (α0 , α, β, γ0 , γ, δ; x) = α0 φ0 (r) +

N ∑


m=1

+γ0 χ0 (r) +

N ∑ m=1


N ∑

βm φm (r) sin mϑ

m=1 N ∑ m=1

δm χm (r) sin mϑ,

x = (x, y) ∈ Ω,

(6.2)


13

where the T-complete system is given by φm (r) = rm

and

χm (r) = r2+m ,

m = 0, . . . , N.

(6.3)

In (6.2) there are 4N + 2 unknowns, namely the coefficients α0 , α = [α1 , . . . , αN ]T , β = [β1 , . . . , βN ]T γ0 , γ = [γ1 , . . . , γN ]T , and δ = [δ1 , . . . , δN ]T . We choose the boundary collocation points as in (2.4) and collocating the boundary conditions (6.1b) yields uN (α0 , α, β, γ0 , γ, δ; xn ) = f1 (xn ),

n = 1, . . . , 2N + 1,

(6.4)

and ∂uN (α0 , α, β, γ0 , γ, δ; xn ) = f2 (xn ), ∂n

n = 1, . . . , 2N + 1.

(6.5)

We thus obtain a (4n + 2) × (4N + 2) system of the form (

B11

B12

B21

B22

)(

x1 x2

)

( =

f1 f2

) ,

(6.6)

where the (2N + 1) × (2N + 1) matrices B11 , B12 , B21 , B22 have the form of matrix A in (2.7) with µ and λn defined as µ11 = φ0 (R), λ11 n = φn (R),

µ12 = χ0 (R), λ12 n = χn (R),

′ µ21 = φ′0 (R), λ21 n = φn (R),

′ µ22 = χ′0 (R), λ22 n = χn (R),

respectively. Also, xℓ , fℓ , ℓ = 1, 2, are defined as in (2.16), with fℓ0 = fℓ (x1 ), fℓ1n = fℓ (x1+n ), fℓ2n = fℓ (xN +1+n ), ℓ = 1, 2; n = 1, . . . , N .

6.1.1. Efficient solution of (6.6). We observe that system (6.6) has exactly the same structure as system (2.16) and can therefore be solved efficiently at a cost of O(N log N ) operations using the algorithm described in Section 4.2.

6.2. Problems in annuli. We now consider the boundary value problem ∆2 u = 0

in Ω,

(6.7a)

subject to the boundary conditions u = f1

and

∂u = f2 ∂n

on

∂Ω1 ,

(6.7b)

u = f3

and

∂u = f4 ∂n

on

∂Ω2 ,

(6.7c)

and

where the domain Ω is an annulus of radii R1 and R2 (R2 > R1 ), corresponding to the boundaries ∂Ω1 and ∂Ω2 , respectively.

14

A. KARAGEORGHIS

In the biharmonic TCM for doubly connected domains, we seek an approximation to the solution (6.7a) as a linear combination of T-complete functions in the form [21, 28] N ∑

uN (α0 , α, β, γ0 , γ, δ, ϵ0 , ϵ, ζ, η0 , η, θ; x) = α0 φ0 (r) +


m=1

+γ0 χ0 (r) +

N ∑


N ∑

N ∑

ηm ωm (r) cos mϑ +

m=1

N ∑

δm χm (r) sin mϑ

m=1

ϵm ψm (r) cos mϑ +

m=1

+η0 ω0 (r) +

βm φm (r) sin mϑ

m=1

m=1

+ϵ0 ψ0 (r) +

N ∑

N ∑

ζm ψm (r) sin mϑ

m=1 N ∑

θm ωm (r) sin mϑ,

x = (x, y) ∈ Ω,

(6.8)

m=1

where the functions φm (r), χm (r) are defined as in (6.3) and ψ0 (r) = log r, 2

ω0 (r) = r log r,

ω1 (r) = r log r,

ψm (r) = r−m ,

m = 1, . . . , N,

2−m

m = 2, . . . , N.

ωm (r) = r

,

(6.9)

In (6.8) there are 8N + 4 unknowns, namely the coefficients α0 , α = [α1 , . . . , αN ]T , β = [β1 , . . . , βN ]T , γ0 , γ = [γ1 , . . . , γN ]T , δ = [δ1 , . . . , δN ]T , ϵ0 , ϵ = [ϵ1 , . . . , ϵN ]T , ζ = [ζ1 , . . . , ζN ]T , η0 , η = [η1 , . . . , ηN ]T and θ = [θ1 , . . . , θN ]T . We choose the boundary collocation points as in (2.14) and collocating the boundary conditions (6.7b)-(6.7c) yields uN (α0 , α, β, γ0 , γ, δ, ϵ0 , ϵ, ζ, η0 , η, θ; xn ) = f1 (xn ),

∂uN (α0 , α, β, γ0 , γ, δ, ϵ0 , ϵ, ζ, η0 , η, θ; xn ) = f2 (xn ), ∂n

uN (α0 , α, β, γ0 , γ, δ, ϵ0 , ϵ, ζ, η0 , η, θ; x2N +1+n ) = f3 (x2N +1+n ) ∂uN (α0 , α, β, γ0 , γ, δ, ϵ0 , ϵ, ζ, η0 , η, θ; x2N +1+n ) = f4 (x2N +1+n ), ∂n

and

for n = 1, . . . , 2N + 1.

(6.10)

From (6.10) we obtain a (8N + 4) × (8N + 4) system of the form 

B11

 B  21   B31 B41

B12

B13

B14

B22

B23

B24

B32

B33

B34

B42

B43

B44



  x1 f1   f2   x2   =    f3 x3   x4 f4

  , 

(6.11)

where the (2N + 1) × 1 vectors xℓ , ℓ = 1, 2, are defined as in (2.16), x3 = [ϵ0 , ϵT , ζ T ], x4 = [η0 , η T , θ T ], fℓT = [fℓ0 , f Tℓ1 , f Tℓ2 ], with f ℓk = [fℓk 1 , fℓk 2 , . . . , fℓk N ]T , ℓ = 1, . . . , 4, k = 1, 2, fℓ0 = fℓ (x1 ), fℓ1n = fℓ (x1+n ), fℓ2n = fℓ (xN +1+n ), f(2+ℓ)0 = f2+ℓ (x(2N +1)+1 ), f(2+ℓ)1 n = f2+ℓ (x(2N +1)+1+n ), f(2+ℓ)2 n = f2+ℓ (x(2N +1)+N +1+n ), ℓ = 1, 2, n = 1, . . . , N . The (2N + 1) × (2N + 1) matrices Bkℓ , k, ℓ = 1, . . . , 4, have the form of matrix A in (2.7) with µ


15

and λn defined as µ11 = φ0 (R1 ), λ11 n = φn (R1 ), 13

µ

= ψ0 (R1 ),

µ21 = φ′0 (R1 ), µ23 = ψ0′ (R1 ), µ31 = φ0 (R2 ), 33

µ

= ψ0 (R2 ),

µ41 = φ′0 (R2 ), µ43 = ψ0′ (R2 ),

λ13 n = ψn (R1 ), 21 λn = φ′n (R1 ), ′ λ23 n = ψn (R1 ), λ31 n = φn (R2 ), λ33 n = ψn (R2 ), 41 λn = φ′n (R2 ), ′ λ43 n = ψn (R2 ),

µ12 = χ0 (R1 ), λ12 n = χn (R1 ), µ14 = ω0 (R1 ), λ14 n = ωn (R1 ), ′ µ22 = χ′0 (R1 ), λ12 n = χn (R1 ), ′ µ24 = ω0′ (R1 ), λ24 n = ωn (R1 ),

µ32 = χ0 (R2 ), λ32 n = χn (R2 ), µ34 = ω0 (R2 ), λ34 n = ωn (R2 ), ′ µ42 = χ′0 (R2 ), λ42 n = χn (R2 ), ′ µ44 = ω0′ (R2 ), λ44 n = ωn (R2 ),

respectively. 6.2.1. Efficient solution of (6.11). System  ˆ B11 Bˆ12  Bˆ  21 Bˆ22   Bˆ31 Bˆ32 Bˆ Bˆ 41

42

(6.11) can be re-written as   ˆ1 x Bˆ13 Bˆ14  ˆ  Bˆ23 Bˆ24   x 2    = ˆ3  Bˆ33 Bˆ34   x ˆ4 x Bˆ Bˆ 43

44



 f1  f2     f3  , f4

(6.12)

where the matrices Bˆkℓ , k, ℓ = 1, . . . , 4, have the form of matrix A in (2.7) with µ and λn defined as ˆ 11 = 1, µ µ ˆ11 = 1, λ ˆ12 = 1, n µ21 ˆ 21 µ22 ˆ 22 λ21 λ22 n n µ ˆ21 = 11 , λ µ ˆ22 = 12 , λ n = 11 , n = 12 , µ λn µ λn µ31 ˆ 31 λ31 µ32 ˆ 32 λ32 n n µ ˆ31 = 11 , λ µ ˆ32 = 12 , λ n = 11 , n = 12 , µ λn µ λn µ42 ˆ 42 µ41 ˆ 41 λ41 λ42 n n µ ˆ42 = 12 , λ µ ˆ41 = 11 , λ n = 11 , n = 12 , µ λn µ λn

ˆ 12 = 1, µ ˆ 13 = 1, µ ˆ 14 = 1, λ ˆ13 = 1, λ ˆ14 = 1, λ n n n µ23 ˆ 23 µ24 ˆ 24 λ23 λ24 n 23 24 µ ˆ = 13 , λn = 13 , µ ˆ = 14 , λn = n14 , µ λn µ λn 33 33 34 34 µ µ ˆ 33 = λn , µ ˆ 34 = λn , µ ˆ33 = 13 , λ ˆ34 = 14 , λ n n 13 µ λn µ λ14 n µ43 ˆ 43 µ44 ˆ 44 λ43 λ44 n 43 44 µ ˆ = 13 , λn = 13 , µ ˆ = 14 , λn = n14 , µ λn µ λn

ˆ = λ11 · β,ˆ ˆ ℓ , ℓ = 1, . . . , 4, we have α ˆ = λ11 · α, β respectively. Also, with the obvious notation for x ˆ 0 = µ11 α0 , α γ0 = 12 12 13 13 14 14 12 13 14 ˆ ˆ ˆ ˆ ˆ ˆ µ γ0 , γ = λ · γ, δ = λ · δ, ϵˆ0 = µ ϵ0 , ϵ = λ · ϵ, ζ = λ · ζ, ηˆ0 = µ η0 , η = λ · η and θ = λ · θ. Since Bˆ11 = Bˆ12 = Bˆ13 = Bˆ14 , system (6.12) can be written as  ˆ     ˜1 x B11 0 0 0 f1  Bˆ  ˆ   f2   21 B˜22 B˜23 B˜24   x 2   (6.13)    =   f3  ,  Bˆ31 B˜32 B˜33 B˜34   x ˆ3  f4 ˆ4 x Bˆ41 B˜42 B˜43 B˜44 ˜1 = x ˆ1 + x ˆ2 + x ˆ3 + x ˆ 4 and the matrices B˜kℓ , k, ℓ = 2, 3, 4 have the form of matrix A in (2.7) with µ and λn where x defined as ˜ 22 = λ ˆ 22 − λ ˆ 21 , µ ˜ 23 = λ ˆ 23 − λ ˆ 21 , µ ˜ 24 = λ ˆ 24 − λ ˆ 21 , µ ˜22 = µ ˆ22 − µ ˆ21 , λ ˜23 = µ ˆ23 − µ ˆ21 , λ ˜24 = µ ˆ24 − µ ˆ21 , λ n n n n n n n n n ˜ 32 = λ ˆ 32 − λ ˆ 31 , µ ˜ 33 = λ ˆ 33 − λ ˆ 31 , µ ˜ 34 = λ ˆ 34 − λ ˆ 31 , µ ˜32 = µ ˆ32 − µ ˆ31 , λ ˜33 = µ ˆ33 − µ ˆ31 , λ ˜34 = µ ˆ34 − µ ˆ31 , λ n

n

n

˜ 42 = λ ˆ 42 − λ ˆ 41 , µ ˜42 = µ ˆ42 − µ ˆ41 , λ n n n

n

n

n

˜ 43 = λ ˆ 43 − λ ˆ 41 , µ ˜43 = µ ˆ43 − µ ˆ41 , λ n n n

n

n

n

˜ 44 = λ ˆ 44 − λ ˆ 41 , µ ˜44 = µ ˆ44 − µ ˆ41 , λ n n n

respectively. System ˜ 1 = f1 Bˆ11 x

(6.14)

16

A. KARAGEORGHIS

can be solved using Algorithm 1 to yield  B˜22  ˜  B 32

B˜42

˜ 1 at a cost of O(N log N ) operations. From system (6.13) we have x     ˆf2 ˆ2 x B˜23 B˜24     ˆ 3  =  ˆf3  , (6.15) B˜33 B˜34   x ˆf ˆ4 x B˜ B˜ 43

44

4

˜ 1 , ℓ = 2, 3, 4. Each of the matrix vector multiplications Bˆℓ1 x ˜ 1 , ℓ = 2, 3, 4 can be carried out where ˆfℓ = fℓ − Bˆℓ1 x using Algorithm 2 at a cost of O(N log N ) operations. System (6.15) can now be written as      ˆf2 ¯2 B¯22 B¯23 B¯24 x    ¯   ¯ 3  =  ˆf3  , (6.16)  B32 B¯33 B¯34   x ¯ ¯ ¯ ˆ ¯4 x B42 B43 B44 f4 where each of the matrices B¯kℓ , k, ℓ = 2, 3, 4 has the form of matrix A in (2.7) with µ and λn defined as ¯ 22 = 1, ¯ 23 = 1, ¯ 24 = 1, µ ¯22 = 1, λ µ ¯23 = 1, λ µ ¯24 = 1, λ n n n 32 33 34 ˜ 32 ˜ 33 ˜ 34 µ ˜ λ µ ˜ λ µ ˜ 33 34 ¯ 32 = n , ¯ 33 = n , ¯ 34 = λn , µ ¯32 = 22 , λ , λ , λ µ ¯ = µ ¯ = n ˜ 22 ˜ 23 ˜ 24 µ ˜ µ ˜23 n µ ˜24 n λ λ λ n n n 43 44 42 ˜ 42 ˜ 43 ˜ 44 λ µ ˜ λ µ ˜ λ µ ˜ ¯ 42 = n , ¯ 43 = n , ¯ 44 = n , µ ¯43 = 23 , λ µ ¯44 = 24 , λ µ ¯42 = 22 , λ n n n ˜ 22 ˜ 23 ˜ 24 µ ˜ µ ˜ µ ˜ λ λ λ n n n ˜ 22 · γ ˜ 22 · δ, ˆ ˜ ℓ , ℓ = 2, 3, 4, we have γ˜0 = µ ˜ = λ ˆ , δ˜ = λ respectively. Also, with the obvious notation for x ˜22 γˆ0 , γ 23 23 24 24 23 24 ˜ ˜ ˜ ˆ ˜ ˜ ˜ ˆ ˜ =λ ·η ˆ and θ = λ · θ. ϵ˜0 = µ ˜ ϵˆ0 , ˜ϵ = λ · ˆϵ, ζ = λ · ζ, η˜0 = µ ˜ ηˆ0 , η Since we have B¯22 = B¯23 = B¯24 , system (6.16) can be written as      ˆf2 ¯2 B¯22 x 0 0   ¯    ˜ 3  =  ˆf3  , (6.17)  B32 B˘33 B˘34   x ¯ ˘ ˘ ˆ ˜4 x B42 B43 B44 f4 ¯2 = x ˜2 + x ˜3 + x ˜ 4 each of the matrices B˘kℓ , k = 3, 4, ℓ = 3, 4 has the form of matrix A in (2.7) with µ and where x λn defined as ¯ 32 , ¯ 34 − λ ˘ 34 = λ ¯ 32 , ¯ 33 − λ ˘ 33 = λ µ ˘34 = µ ¯34 − µ ¯32 , λ µ ˘33 = µ ¯33 − µ ¯32 , λ n n n n n n 43 43 42 ˘ 43 43 42 44 44 42 ˘ 44 44 ¯ ¯ ¯ ¯ µ ˘ =µ ¯ −µ ¯ , λ =λ −λ , µ ˘ =µ ¯ −µ ¯ , λ = λ − λ42 , n

n

n

n

n

n

respectively. We can solve system ¯ 2 = ˆf2 B¯22 x using Algorithm 1 at a cost of O(N log N ) operations. Then we have ( )( ) ( ) ˜f3 ˜3 x B˘33 B˘34 = , ˜f ˜4 x B˘43 B˘44 4

(6.18)

(6.19)

¯ 2 , ℓ = 1, 2. The matrix-vector multiplications B¯ℓ2 x ¯ 2 , ℓ = 1, 2 can be carried out at a cost of where ˜fℓ = ˆfℓ − B¯ℓ2 x O(N log N ) operations using Algorithm 2. System (6.19) is of the form of system (2.16) and can thus be solved using the algorithm described in Section 4.2 at ˜3, x ˜ 4 . From these we can obtain x ˜2 = x ¯2 −x ˜3 −x ˜ 4 . The vectors a cost of O(N log N ) operations to yield the vectors x 22 22 22 23 23 ˆ ˜ ˆ2, x ˆ3, x ˆ 4 are calculated from γˆ0 = γ˜0 /˜ x µ , γˆn = γñ /˜ µ , δn = δn /˜ µ , ϵˆ0 = ϵ˜0 /˜ µ , ϵˆn = ϵñ /˜ µ , ζˆn = ζñ /˜ µ23 , 24 24 24 ˆ ˜ ˆ1 = x ˜1 − x ˆ2 − x ˆ3 − x ˆ 4 . Finally, the µ , ηˆn = ηñ /˜ µ and θn = θn /˜ µ , n = 1, . . . , N . Then we calculate x ηˆ0 = η˜0 /˜ solution xℓ , ℓ = 1, . . . , 4 is calculated from α0 = α ˆ 0 /µ11 , αn = α ˆ n /µ11 , βn = βˆn /µ11 , γ0 = γˆ0 /µ12 , γn = γˆn /µ12 , δn = δˆn /µ12 , ϵ0 = ϵˆ0 /µ13 , ϵn = ϵˆn /µ13 , ζn = ζˆn /µ13 , η0 = ηˆ0 /µ14 , ηn = ηˆn /µ14 and θn = θˆn /µ14 , n = 1, . . . , N .


17

6.3. Example 7. We consider problem (6.1) in a disk of radius R = 2 with boundary conditions corresponding to the exact solution u(x, y) = x sin(2x) cosh(2y). (6.20) We calculated the maximum absolute error EN in the approximation uN for various values of N as in Example 4. In Figure 7 we present the maximum absolute error EN versus N . Error vs N

2

10

0

10

−2

Error

10

−4

10

−6

10

−8

10

4

6

8

10

12 N

14

16

18

20


6.4. Example 8. We finally consider problem (6.7) in an annulus of radii R1 = 1/2, R2 = 2 with the boundary conditions corresponding to the exact solution (6.20). We calculated the maximum absolute error EN in the approximation uN for various values of N as in Example 5. In Figure 8 we present the maximum absolute error EN versus N .

7. Conclusions We considered the efficient solution of the systems resulting from the TCM discretization of harmonic, Helmholtz and modified Helmholtz problems in disks and annular domains. The matrices involved possess certain structures which are exploited to yield efficient algorithms for their solution which make use of FFTs and the ShermanMorrison-Woodbury formula. The numerical results we obtained reveal the high accuracy and rapid convergence of the approximate TCM solutions to the exact ones. The ideas developed for the efficient solution of harmonic, Helmholtz and modified Helmholtz problems in disks and annular domains are shown to be readily applicable to the corresponding biharmonic problems. The application of efficient algorithms to three-dimensional problems using the appropriate T-complete functions [21, pages 32-33] is currently under investigation.

18

A. KARAGEORGHIS

Error vs N

2

10

0

10

−2

Error

10

−4

10

−6

10

−8

10

4

6

8

10

12 N

14

16

18

20

Figure 8. Example 8: Maximum absolute error versus N . Acknowledgements The author is grateful to the University of Cyprus for supporting this research, to Professor Graeme Fairweather for fruitful conversations and to Professor Michael Ng of Hong Kong Baptist University for suggesting the solution of system (3.13) using the Sherman-Morrison-Woodbury formula. The constructive comments of the two anonymous referees are also gratefully acknowledged.

Appendix ˆ The following lemma proves the properties of matrices Aˆ and B. ˆ defined in (3.2) satisfy the identities Lemma. For every N ∈ N∗ , the matrices Aˆ and B ˆ 2 = 2N + 1 IN B 4 and

(

Aˆ − H

)(

) 1 N 1 Aˆ − H = (N + )H + ( + )IN . 2 2 4

ˆ 2 are given by Proof. The diagonal elements bn , n = 1, . . . , N of the matrix B ( ) N ∑ 2πn 2 bn = sin m , 2N + 1 m=1 or, using [14, 1.351] bn =

[ ( ) ] ( ) 1 2πn 2πn 1 (2N + 1) sin − sin (2πn) csc = (2N + 1). 4 2N + 1 2N + 1 4

(A.1)

(A.2)


19

ˆ 2 are given by The off-diagonal elements bn1 n2 , n1 , n2 = 1, . . . , N, n1 ̸= n2 of the matrix B ( ) ( ) ( ) ( ) N N N ∑ 2πn1 2πn2 1 ∑ 2π(n1 − n2 ) 1 ∑ 2π(n1 + n2 ) bn1 n2 = sin m sin m = cos m − cos m . 2N + 1 2N + 1 2 m=1 2N + 1 2 m=1 2N + 1 m=1 (A.3) From [14, 1.342] ( ) ) ) ) ( ( ( N ∑ 2π(n1 − n2 ) π(n1 − n2 ) π(n1 − n2 ) π(n1 − n2 ) cos m = cos (N + 1) sin N csc 2N + 1 2N + 1 2N + 1 2N + 1 m=1 [ ( )] ( ) 1 π(n1 − n2 ) π(n1 − n2 ) 1 = (A.4) sin (π(n1 − n2 )) − sin csc =− 2 2N + 1 2N + 1 2 and similarly ( ) ( ) ( ) ( ) N ∑ 2π(n1 + n2 ) π(n1 + n2 ) π(n1 + n2 ) π(n1 + n2 ) cos m = cos (N + 1) sin N csc 2N + 1 2N + 1 2N + 1 2N + 1 m=1 [ ( )] ( ) 1 π(n1 + n2 ) π(n1 + n2 ) 1 = sin (π(n1 + n2 )) − sin csc =− , (A.5) 2 2N + 1 2N + 1 2 which from (A.3) yield that bn1 n2 = 0, n1 , n2 = 1, . . . , N, n1 ̸= n2 . Regarding (A.2), we can write

(

Aˆ − H

)(

) ˆ + H 2. Aˆ − H = Aˆ2 − H Aˆ − AH

(A.6)

The diagonal elements an , n = 1, . . . , N of the matrix Aˆ2 are given by ) ( N ∑ 2πn 2 an = , cos m 2N + 1 m=1 or, using [14, 1.351]

( ) ( ) ( ) N 1 2πn 2πn 2πn an = + cos (N + 1) sin N csc 2 2 2N + 1 2N + 1 2N + 1 [ ( )] ( ) N 1 2πn 2πn N 1 2N − 1 = + sin (2πn) − sin csc = − = . 2 4 2N + 1 2N + 1 2 4 4 The off-diagonal elements an1 n2 , n1 , n2 = 1, . . . , N, n1 ̸= n2 of the matrix Aˆ2 are given by ( ) ( ) ( ) ( ) N N N ∑ 2πn1 2πn2 1 ∑ 2π(n1 − n2 ) 1 ∑ 2π(n1 + n2 ) an1 n2 = cos m cos m = cos m + cos m , 2N + 1 2N + 1 2 m=1 2N + 1 2 m=1 2N + 1 m=1 (A.7) which from (A.4)-(A.5) yields that 1 an1 n2 = − . 2 Therefore we can write 2N + 1 1 Aˆ2 = IN − H. (A.8) 4 2 ˆ and H Aˆ are equal. If we consider H A, ˆ the Due to the structures of the matrices Aˆ and H the matrices AH th elements of its n column are all equal to ) ( N ∑ 2πn1 , n = 1, . . . N, cn = cos m 2N + 1 m=1

20

or using [14, 1.342] again (cf. (A.4)) ( cn = cos (N + 1)

A. KARAGEORGHIS

πn 2N + 1

)

( sin N

πn 2N + 1

)

( csc

πn 2N + 1

)

1 =− . 2

Therefore ˆ = H Aˆ = − 1 H. AH 2

(A.9)

H 2 = N H.

(A.10)

Finally, Substitution of the expressions (A.8), (A.9) and (A.10) into (A.6) yields the desired result (A.2).

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Kuproc

Panepisthmio Kuprou, P.O.Box 20537, 1678 Nicosia/Leukwsia,