6 Domain decomposition and fictitious domain methods with distributed Lagrange multipliers

Transcription

1 M _ Thirteenth International Conference on Domain Decomposition Methos Eitors:. Debit M.Garbey R. Hoppe J. Périaux D. eyes Y. uznetsov c 2 DDM.org Domain ecomposition an fictitious omain methos with istribute agrange multipliers Yu.A. uznetsov Introuction In this paper we consier three applications of the istribute agrange multiplier technique [DGH 92 GHJ 97 G98] to esign new omain ecomposition an fictitious omain methos for the iffusion equation () in a boune 2D/3D polygonal omain with the homogeneous Dirichlet bounary conition (2) an a piece-wise constant iffusion coefficient. The above restrictions are impose for the sake of simplicity. The generalizations of the algorithms an theoretical results to more complicate equations omains an bounary conitions are obvious. et be a triangular/tetraheral partitioning of an! be the corresponing piecewise linear finite element subspace of "$. We shall always assume in this paper that is a shape-regular mesh. Then the classical finite element metho ' (. ) /! ' () *+ () )-! (3) )7 an *8 () 9 results in the system of linear algebraic equations = )/ : (4) with a symmetric positive efinite matrix >@?BA? C D EGF! an a vector HI>J?. We also enote by the mass matrix an by the lumpe mass matrix i.e. is iagonal an M M5 O QPRTUTVUPW M >J?. For an Y being non-overlapping subomains of such that D /[\J Y we enote by an the corresponing stiffness matrices an by an ] U the corresponing mass (lumpe mass) matrices. The matrices an can be introuce by subassembling of matrices with the same subassembling matrices _ ` ap]cb respectively. For instance _ Department of Mathematics University of Houston Houston T UA kuz@math.uh.eu

2 : UETOV Domain ecomposition for composite materials et be a rectangle an ` P] P be open non-overlapping polygonal subomains of i.e. [ for ` an ` PR. An example of is given in Figure. We assume that are shape-regular E]FW an E! "$R B' with some positive constants an $ (' is given. We also assume that OP )+* -.5C/ $ VP$2 in ` P] an - P in the rest of. We shall call this moel example a composite material. Figure : The computational gri. an The stiffness matrix Y ) ) of system (4) can be presente in the form * 8 4 P 7 3)] 4W 8 )] 8 =! 3) 4U 8 ) 8! It is obvious that with an appropriate permutation matrix : : >= we have (5)

3 ' 3 ` DOMAI DECOMPOITIO AD FICTITIOU DOMAI METHOD 9 is the stiffness matrix of the aplacian for the subomain P. In [uz] was propose to replace system (4) with in (5) by a sale point system with an the block iagonal matrix O ITT >?BA 3? () > 3?8A 3? to (4) coin- ystem () is equivalent to system (4) in the sense that the solution vector cies with the solution subvector to () an vice versa. Moreover $ for any solution subvector to () ` P]. et a matrix " " be spectrally equivalent to i.e " )' ) )' )2 " )B )2 )=>? with positive constants an inepenent of the Then the matrix with "! E $ P " "! TT (7) enotes the generalize inverse to ` PR was propose in [uz] as an effective preconitioner for the matrix in (). To justify the latter statement we have to consier the matrix in its invariant subspace ` supplie with the scalar prouct generate by the matrix '! E($ " '! VUTV 3 3 It can be easily shown that is a symmetric operator in ` with respect to the scalar prouct. Moreover ` `. To this en all non-zero eigenvalues of the matrix belong to the union of two segments ) 2 an +* ) * 2 with en points - -

4 5 3 - F F 7 UETOV The conition number of with respect to the subspace ` efine by ($ * E $ Uner all the above assumptions the following result was prove in [uz]. * an the -scalar prouct is Proposition (8) is a positive constant inepenent of the values Remark In general the constant epens on the constants The implementation proceure of the preconitioner that VUT 3 an the mesh. ` P]. is base on a simple observation (9) The results of numerical experiments for the geometry given in Fig. are presente in Table. For numerical experiments " was chosen to be the BP-preconitioner [BP9]. PU PU PU PU Table. The number of PCG iterations. PHP PR$PR P The vectors ` PR in () can be calle the iscrete istribute agrange multipliers. They have a very simple connection with the continuous/ifferential istribute agrange multiplier. ystem () can be obtaine by the straightforwar finite element iscretization of the variational problem: fin "$ "$ ` P] such that )= "H 9 * "H 3-4U3)J 3-4U - ` PR. 4 9 * 9 * 4W)J 45 )/ ` PR$ ()

5 P 9 DOMAI DECOMPOITIO AD FICTITIOU DOMAI METHOD 7 Fictitious omain metho The name fictitious omain metho was originally suggeste by V.. aul ev in [au3]. The aul ev s iea is to replace ifferential problem () (2) by the problem ) 3 ) ) ) () is a rectangle containing the original simply-connecte omain ) It was prove that ) ) as ) : I I The form of the equation in () remins us the situation consiere in the previous section. If we introuce the istribute agrange multiplier by in P (2) " then the weak sale point formulation reas as follows: fin D such that "H on 9 3 4U3)7 3 4U - 9 4W3)7 4U ) )7 : )= "H $ "H on. The interesting observation is that with formulation (3) coincies with the istribute agrange multiplier fictitious omain metho invente by R. Glowinski (see [DGH 92 GHJ 97]). Thus the Glowinski s metho is the closure with respect to the parameter of the aul ev s metho. The finite element iscretization to (3) results in the algebraic system c - c c (4) 8 8 c stays for the stiffness matrix in subomain an c 8 = (3)

6 * ' ' * 72 UETOV stays for the stiffness matrix in the rectangle. If we present in a ifferent block form: = an assume that a matrix " is spectrally equivalent to can be propose in the form of the block iagonal matrix c then the preconitioner for " "! = (5) "! c. Assume that the norm preserving finite element extension theorem for the subomain with respect to the rectangle hols. Then is a positive constant inepenent of the mesh an value of ) P2. In the case the result was prove in [G98]. For the case one has to use technique from [uz]. Overlapping omain ecomposition et be partitione into two subomains an such that is nonempty. We assume that F 7 U $.5C/$ an the norm preserving finite element extension results from into an hol [Wi87]. ater we shall give the algebraic interpretation of this assumption. et the bilinear form ' (. ) be split into two bilinear forms [uz97]: ' (. ) / (. )2 an the linear form *8 () be also splitte into two linear forms: with an with *8 () * (.) D * () :) () * () (7) 3-4W3)7 B ]b )27 * )7 P] P ]b

7 " DOMAI DECOMPOITIO AD FICTITIOU DOMAI METHOD 73 ` OP]Yb. Then let us efine two new bilinear an linear forms by ) ) ) )! : *Q ) ) ) * )7 )- " ) ( ) 45 ) * ] ) W 2] (' ) U $) W :) on an!(! " on ` OP]cb Then the weak formulation of () base on the above overlapping ecomposition with istribute agrange multipliers can be given by: fin!!!!! such that )=! H!!. : )2 ) *Q (). The finite element iscretization of (9) can be suggeste with the same formulae by replacing! an!! by! an!! which are the traces of the finite element space! onto an respectively. The finite element iscretization of (9) results in the system of algebraic equations Here is efine by = - 8 = (8) (9) ' (2) c = 4U :!! (2) i.e. is the stiffness matrix for the aplacian in the subomain. We introuce a preconitioner for in the form of a block iagonal matrix: " "! (22)

8 " 74 UETOV " is spectrally equivalent to ` P]Yb an "! is spectrally equivalent to the chur complement matrix! (23) We have plenty of choices for " an " for instance multigri preconitioner. The question is only about a choice for "!. The assumption about the norm preserving finite element extension results (in the context of the above metho) is equivalent to the assumption that the matrix is spectrally equivalent to matrices G ` ap]yb In this case simple transformations show that the matrix matrix. The conclusion is obvious: we have to choose Implementation proceure for " "!! is very simple ue to the formulae "! O! an a! '! is spectrally equivalent to the Proposition 2 Uner the assumptions mae the eigenvalues of the matrix the union of two segments ) 2 * ) * 2 with the en points inepenent of the mesh. Remark 2 The values of spectral equivalence " an as well as belong to an from Proposition 2 epen on the constants of an ` ap]cb. Acknowlegments: This work was partially supporte by F (grant CCR-99235) an by os Alamos Computer cience Institute (ACI). The author thanks.ipnikov for proviing the numerical experiments an technical assistance. References [BP9]James H. Bramble Joseph E. Pasciak an Jinchao u. Parallel multilevel preconitioners. Math. Comp. 55: [DGH 92]Q. V. Dihn R. Glowinski J. He V. wock T. W. Pan an J. Périaux. agrange multiplier approach to fictitious omain methos: Application to flui ynamics an electromagnetics. In Davi E. eyes Tony F. Chan Gérar A. Meurant Jeffrey. croggs an Robert G. Voigt eitors Fifth International ymposium on Domain Decomposition Methos for Partial Differential Equations pages 5 94 Philaelphia PA 992. IAM.

9 DOMAI DECOMPOITIO AD FICTITIOU DOMAI METHOD 75 [GHJ 97]R. Glowinski T. I. Hesla D.D. Joseph T.W. Pan an J. Periaux. Distribute agrange multiplier methos for particulate flows. In M.O. Bristeau G.J. Etgen W. Fitzgibbon J.. ions J. Periaux an M.F. Wheeler eitors Computational cience for the 2st Century pages Chichester 997. Wiley. [G98]Rolan Glowinski an Yuri uznetsov. On the solution of the Dirichlet problem for linear elliptic operators by a istribute agrange multiplier metho. C. R. Aca. ci. Paris ér. I Math. 327(7): [uz97]yuri. A. uznetsov. Overlapping omain ecomposition with non matching gris. In Petter E. Bjørsta Magne Espeal an Davi eyes eitors Domain Decomposition Methos in ciences an Engineering. J. Wiley 997. Proceeings from the inth International Conference June 99 Bergen orway. [uz]yuri A. uznetsov. ew iterative methos for singular perturbe positive efinite matrices. Russian J. umer. Anal. Math. Moelling 5: [au3]valerij. aul ev. On solution of some bounary value problems on high performance computers by fictitious omain metho. iberian Math. J. 4(4): (in Russian). [Wi87]Olof B. Wilun. An extension theorem for finite element spaces with three applications. In Wolfgang Hackbusch an ristian Witsch eitors umerical Techniques in Continuum Mechanics pages 22 Braunschweig/Wiesbaen 987. otes on umerical Flui Mechanics v. Frier. Vieweg un ohn. Proceeings of the econ GAMM- eminar iel January 98.