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A Matrix Decomposition MFS Algorithm for Problems in Hollow Axisymmetric Domains

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A Matrix Decomposition MFS Algorithm for Problems in Hollow Axisymmetric Domains
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  DOI: 10.1007/s10915-005-9006-3 Journal of Scientific Computing, Vol. 28, No. 1, July 2006 (© 2005) A Matrix Decomposition MFS Algorithmfor Problems in Hollow Axisymmetric Domains Th. Tsangaris, 1 Y.-S. Smyrlis, 1 and A. Karageorghis 1 Received February 23, 2004; accepted (in revised from) July 10, 2005; Published online December 7, 2005 In this work we apply the Method of Fundamental Solutions (MFS) withfixed singularities and boundary collocation to certain axisymmetric harmonicand biharmonic problems. By exploiting the block circulant structure of thecoefficient matrices appearing when the MFS is applied to such problems, wedevelop efficient matrix decomposition algorithms for their solution. The algo-rithms are tested on several examples. KEY WORDS:  Method of fundamental solutions; Laplace equation; biharmon-ic equation; hollow axisymmetric domains; circulant matrices; fast Fourier trans-form. AMS SUBJECT CLASSIFICATION:  Primary 65N12; 65N38; Secondary65N15; 65T50; 35J25. 1. INTRODUCTION In this paper, we investigate the application of the Method of Fundamen-tal Solutions (MFS) to certain axisymmetric harmonic and biharmonicproblems. In particular, we consider the MFS with fixed singularities forharmonic and biharmonic problems in axisymmetric hollow domains. Weextend the ideas developed in [14], where the MFS is applied to har-monic problems in axisymmetric simply-connected domains, and [8], wherethe MFS is applied to the corresponding biharmonic problems. In theproblems examined in this study, the MFS discretization leads to lin-ear systems the coefficient matrices of which have block circulant struc-tures. Matrix decomposition algorithms are developed for the efficient 1 Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678Nicosia, Cyprus/ K  ´ υπ̺oς  . E-mails:  { mspgtt1,smyrlis,andreask } @ucy.ac.cy 31 0885-7474 / 06 / 0700-0031 / 0 © 2005 Springer Science+Business Media, Inc.  32 Tsangaris, Smyrlis, and Karageorghis solution of these systems. These algorithms also make use of fast Fouriertransforms (FFT). Comprehensive reviews of the recent developments andapplications of the MFS and related methods may be found in the sur-vey papers [2, 6, 7, 10]. Also, the books [4, 9, 11] provide useful infor-mation concerning various implementational and theoretical aspects of theMFS. In domain–discretization techniques such as finite element and finitedifference methods, the reduction of the three-dimensional axisymmetricproblem to a two-dimensional problem governed by the axisymmetric ver-sion of the governing equation is important because of the complica-tions involved in the discretization of three-dimensional domains [5]. Thisdifficulty is not as pronounced in the MFS as it is a meshless method.Further, the fundamental solutions of these (two-dimensional) equationsare complicated and involve complete elliptic integrals. Finally, when theboundary conditions of the problem are not axisymmetric, the two-dimen-sional approach requires the solution of a sequence of boundary valueproblems. The approach that we are suggesting in this study avoids thesecomplications. 2. THE HARMONIC CASE2.1. MFS Formulation We consider the three-dimensional boundary value problem ∆u  =  0 in  Ω,u  =  f   on  ∂Ω 1 ,u  =  g  on  ∂Ω 2 , where  ∆  denotes the Laplace operator and  f   is a given function. Theregion  Ω ⊂ R 3 is axisymmetric, which means that it is formed by rotatinga region  Ω ′ ∈ R 2 about the  z -axis. The boundary of   Ω  is  ∂Ω = ∂Ω 1 ∪ ∂Ω 2 and the boundary of   Ω ′ is defined by the two boundary segments  ∂Ω ′ 1 and  ∂Ω ′ 2 , which generate  ∂Ω 1  and  ∂Ω 2 , respectively. In the MFS [6, 14],the solution  u  is approximated by u MN  ( c , d  , R , S  ; P)  = M   m = 1 N   n = 1 c m,n k 1 (P,R m,n ) + M   m = 1 N   n = 1 d  m,n k 1 (P,S  m,n ), P   ∈ Ω, where  c = (c 11 ,c 12 ,... ,c 1 N  ,... ,c M  1 ,... ,c MN  ) T  ,  d  = (d  11 ,d  12 ,... ,d  1 N  ,...,d  M  1 ,... ,d  MN  ) T  and  R ,  S   are 3 MN  -vectors containing the coordinates  A Matrix Decomposition MFS Algorithm 33 of the singularities (sources)  R m,n , S  m,n , m  =  1 ,... ,M, n  =  1 ,... ,N  ,which lie outside  Ω . The function  k 1 (P,R)  is the fundamental solution of Laplace’s equation in  R 3 given by k 1 (P,R)  =  14 π  | P  − R | with  | P   − R |  denoting the distance between the points  P   and  R . The sin-gularities  R m,n , S  m,n  are fixed on the boundary  ∂  ˜ Ω = ∂  ˜ Ω 1 ∪ ∂  ˜ Ω 2  of a solid ˜ Ω  surrounding  Ω . The solid  ˜ Ω  is generated by the rotation of the pla-nar domain  ˜ Ω ′ which is similar to  Ω ′ . Clearly  ∂  ˜ Ω 1  and  ∂  ˜ Ω 2  are simi-lar to  ∂Ω 1  and  ∂Ω 2 , respectively. Also, the boundary of   ˜ Ω ′ is defined bythe segments  ∂  ˜ Ω ′ 1  and  ∂  ˜ Ω ′ 2 , which generate  ∂  ˜ Ω 1  and  ∂  ˜ Ω 2 , respectively.A set of   MN   collocation points  { P  i,j  } M,N i = 1 ,j  = 1  is chosen on  ∂Ω 1  and aset of   MN   collocation points  { Q i,j  } M,N i = 1 ,j  = 1  is chosen on  ∂Ω 2  in the fol-lowing way: We first choose  N   points  { P  j  } N j  = 1  on the boundary segment ∂Ω ′ 1  and  N   points  { Q j  } N j  = 1  on  ∂Ω ′ 2 . These can be described by their polarcoordinates  (r P  j  , z P  j  ), (r Q j  , z Q j  ), j   =  1 ,... ,N  , where  r P  j  , r Q j   denotesthe vertical distance of the points  P  j  , Q j   from the  z -axis and  z P  j  , z Q j  denotes the  z -coordinate of the points  P  j  , Q j  -respectively. The points on ∂Ω 1  are, taken to be x P  i,j   = r P  j   cos  ϕ i , y P  i,j   = r P  j   sin  ϕ i , z P  i,j   = z P  j  and the points on  ∂Ω 2  are x Q i,j   = r Q j   cos  ϕ i , y Q i,j   = r Q j   sin  ϕ i , z Q i,j   = z Q j  , where  ϕ i  =  2 (i  −  1 )π/M, i  =  1 ,... ,M   . Similarly, we choose a set of  MN   singularities  { R m,n } M,N m = 1 ,n = 1  on  ∂  ˜ Ω 1  and a set of   MN   singulari-ties  { S  m,n } M,N m = 1 ,n = 1  on  ∂  ˜ Ω 2  by taking  R m,n  =  (x R m,n ,y R m,n ,z R m,n ), S  m,n  = (x S  m,n ,y S  m,n ,z S  m,n ) , and x R m,n  =  r R n  cos  ψ m , y Q m,n  = r R n  sin  ψ m , z R m,n  = z R n ,x S  m,n  =  r S  n  cos  ψ m , y S  m,n  = r S  n  sin  ψ m , z S  m,n  = z S  n , where  ψ i  = 2 (α + i  − 1 )π/M, i  = 1 ,... ,M  . The parameter  α ∈ [ − 1 / 2 , 1 / 2]describes the rotation of the singularities in the azimuthal direction. The N   points  R j   are chosen on  ∂  ˜ Ω ′ 1  whereas the  N   points  S  j   are chosen on ∂  ˜ Ω ′ 2 . The coefficients  c  and  d   are determined so that the boundary condi-tion is satisfied at the boundary points  { P  i,j  } M,N i = 1 ,j  = 1 ,  { Q i,j  } M,N i = 1 ,j  = 1 : u MN  ( c , d  , R , S  ; P  i,j  ) = f  1 (P  i,j  ), u MN  ( c , d  , R , S  ; Q i,j  ) = f  2 (P  i,j  ),  34 Tsangaris, Smyrlis, and Karageorghis i  = 1 ,... ,M, j   = 1 ,... ,N.  This yields an 2 MN   × 2 MN   linear system of the form  A BC D  cd    =  f g  ,  (2.1)where the matrices  A, B, C  and  D  are block circulant [3]  MN   × MN  matrices, that is A = circ (A 1 ,A 2 ,... ,A N  ), B = circ (B 1 ,B 2 ,... ,B N  ),C = circ (C 1 ,C 2 ,... ,C N  ), D = circ (D 1 ,D 2 ,... ,D N  ). The matrices  A ℓ ,B ℓ ,C ℓ ,D ℓ ,ℓ = 1 ,... ,M  , are  N   × N   matrices defined by (A ℓ ) j,n =  14 π | P  1 ,j   − R ℓ,n | , (B ℓ ) j,n =  14 π | P  1 ,j   − S  ℓ,n | ,(C ℓ ) j,n =  14 π | Q 1 ,j   − R ℓ,n | , (D ℓ ) j,n =  14 π | Q 1 ,j   − S  ℓ,n | ,ℓ = 1 ,... ,M j, n = 1 ,... ,N.  The system (2.1) can therefore be written as   M   ℓ = 1 P  ℓ − 1 ⊗ A ℓ  c +   M   ℓ = 1 P  ℓ − 1 ⊗ B ℓ  d   =  f  ,   M   ℓ = 1 P  ℓ − 1 ⊗ C ℓ  c +   M   ℓ = 1 P  ℓ − 1 ⊗ D ℓ  d   =  g , where the matrix P   is the  M  × M   permutation matrix  P  = circ ( 0 , 1 , 0 ,... , 0 ) and ⊗ denotes the matrix tensor product [12]. 2.2. Matrix Decomposition Algorithm In the case we are examining, a Matrix Decomposition Algorithm [1]involves the reduction of the 2 MN   × 2 MN   system (2.1) to  M   decoupled2 N   × 2 N   systems. This is achieved by exploiting the block circulant struc-ture of the matrices  A, B ,  C  and  D . If   U   is the unitary  M   × M   Fouriermatrix, it is well-known [3, 13] that circulant matrices are diagonalized inthe following way. If   C  = circ (c 1 ,... ,c M  ) , then  C  = U  ∗ DU,  where  D  = diag ( ˆ c 1 ,... ,  ˆ c M  ) , and ˆ c j   = M   k = 1 c k  ω (k − 1 )(j  − 1 ) .  A Matrix Decomposition MFS Algorithm 35 In particular, the permutation matrix  P  = circ ( 0 , 1 , 0 ,... , 0 )  is diagonal-ized as  P  = U  ∗ TU,  where T   =  diag  1 ,ω,ω 2 ,... ,ω M  − 1  , ω  =  e 2 π i /M  .  (2.2)Next we simplify system (2.1). Let  ˜ A  ˜ B ˜ C  ˜ D   =  U   ⊗  I  N   00  U   ⊗  I  N   A BC D  U  ∗ ⊗  I  N   00  U  ∗ ⊗  I  N   and  ˜ c ˜ d   =  U   ⊗  I  N   00  U   ⊗  I  N   cd   ,  ˜ f  ˜ g   =  U   ⊗  I  N   00  U   ⊗  I  N   f g  . Then, after pre-multiplication by  I  2 ⊗ U   ⊗  I  N  , (2.1) becomes  ˜ A  ˜ B ˜ C  ˜ D  ˜ c ˜ d   =  ˜ f  ˜ g  .  (2.3)Since A  = M   k = 1 P  k − 1 ⊗ A k , then ˜ A  =  (U   ⊗  I  N  )   M   k = 1 P  k − 1 ⊗ A k  (U  ∗ ⊗  I  N  ) = M   k = 1  U  P  k − 1 U  ∗  ⊗ A k  = M   k = 1 T  k − 1 ⊗ A k and similarly ˜ B  = M   k = 1 T  k − 1 ⊗ B k ,  ˜ C  = M   k = 1 T  k − 1 ⊗ C k  and  ˜ D  = M   k = 1 T  k − 1 ⊗ D k .
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