Glowinski, R. and O. Pironneau 1979 - Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem, SIAM Rev. 21 (2), 167-212  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The number of unknown constants C k is the same as before. The boundary conditions for component k are as before, but with added Neumann condition = 0 on each solid boundary. Eq. 9) then becomes A i n It is less easy to show that the solution constructed this way is unique. In [3], Glowinski and Pironneau also studied uniqueness of two dimensional flow problems. Their set of equations is different from ours. Moreover, their technique to prove uniqueness is not applicable to a non self adjoint problem, which we have due to the presence of Coriolis terms. 4 Discretization ....

Glowinski, R. and O. Pironneau 1979 - Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem, SIAM Rev. 21 (2), 167-212

.... le cas discret, a un nombre fini de probl emes de Dirichlet scalaires ind ependantes plus un probl eme lineaire pour une inconnue scalaire d efinie sur la fronti ere [7] La base de cette m ethode est la technique de d ecouplage appel ee matrice d influence propos ee par Glowinski et Pironneau in [5] et adopt ee souvent dans la litt erature (voire par example [1] 7] 8] 9] et les r ef erences) Dans cette Note on donne la formulation variationnelle du probl eme de Poisson consid er e et on d emontre l existence et l unicit e de sa solution dans les espaces convenables. Ensuite une ....

....for the surface linear operator introduced in [7] The same argument applies to the right hand side of the linear problem. Subproblem (18a) is a classical vector Dirichlet problem and is easily solved as D independent scalar Dirichlet problems. Let us now consider subproblem (18b) Similarly to [5], 1] and [9] we introduce an isomorphism between H 1=2 ( Gamma) and H 1 T( Omega Gamma3 ae Gamma u H (ae) defined by: 8 ae 2 H 1=2 ( Gamma) u H (ae) is the unique solution of the following classical vector Dirichlet problem: 8 : Find u H (ae) 2 H 1 T ( Omega Gamma such ....

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R. Glowinski and O. Pironneau. Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem, SIAM Review 21, 1979, p.167--212.

....arises in fluid mechanics, in which case OE and represent the streamfunction and vorticity, respectively. It is also a model for plate bending, in which case OE is the deflection and is the bending moment. Our theoretical setting is that established in the review paper of Glowinski Pironneau [8]. We review the essential features here. The standard mixed approximation method for (1) is as follows: we seek ( OE) 2 H 1 ( Omega Gamma Theta H 1 0 ( Omega Gamma satisfying ( AE) Gamma (rAE; rOE) 0 8AE 2 H 1( Omega Gamma Gamma(r ; r ) Gamma(f; 8 2 H 1 0( Omega Gamma ; ....

....preconditioner for the Schur complement in (8) namely S = BM Gamma1 B t . Our objective here is to explore the use of black box multigrid approximations, designed for Dirichlet Poisson problems, as a preconditioner for (8) To see how such a preconditioner might be defined, we follow [8] and [2] by decomposing h into the sum of interior and boundary contributions: n I nB X i=1 i i z h 2X h = n I X j=1 v j j z v h 2M h nB X k=1 k n I k z h 2T h ; 13) where T h L 2 ( Omega Gamma1 This decomposition induces a ....

R. Glowinski and O. Pironneau. Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev., 21:167--212, 1979.

....a condition number which grows at least like O( H=h) 2 ) Better algorithms 3 are obtained by using certain vertex spaces. Some parallel multilevel algorithms are also constructed for the biharmonic problem. Some earlier works on the biharmonic problem can be found in Glowinski and Pironneau [30], Bjrstad [2] Widlund [45] and Chan, E and Sun [18] The thesis is organized as follows. In the remainder of chapter 1, we review some basic Sobolev spaces. We also discuss some preliminary material for the biharmonic problem and some standard finite element discretizations. In chapter 2, we ....

R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional stokes problem, SIAM SiRev., 21 (1979), pp. 167-- 212.

....before. The boundary conditions for component k are as before, but with added Neumann condition n = 0 on each solid boundary. Eq. 9) then becomes I Gamma k A i n ds = Gamma I Gamma k F Delta sds (13) It is less easy to show that the solution constructed this way is unique. In [3], Glowinski and Pironneau also studied uniqueness of two dimensional flow problems. Their set of equations is different from ours. Moreover, their technique to prove uniqueness is not applicable to a non self adjoint problem, which we have due to the presence of Coriolis terms. 4 Discretization ....

Glowinski, R. and O. Pironneau 1979 - Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem, SIAM Rev. 21 (2), 167-212

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