Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

• Published in: Mathematics of computation 2009-01, Vol.78 (266), p.789-814
• Main Authors: Dolean, Victorita, Nataf, Frédéric, Rapin, Gerd
• Format: Article
• Language: English
• Subjects: Airy equation; Boundary conditions; Coefficients; Decomposition methods; Differential operators; Exact sciences and technology; Factorization; Fourier transformations; Mathematical analysis; Mathematical vectors; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical Analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Polynomials; Scalars; Sciences and techniques of general use
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/S0025-5718-08-02172-8

In this paper the Smith factorization is used systematically to derive a new domain decomposition method for the Stokes problem. In two dimensions the key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show, how a proposed domain decomposition method for the bi-harmonic problem leads to a domain decomposition method for the Stokes equations which inherits the convergence behavior of the scalar problem. Thus, it is sufficient to study the convergence of the scalar algorithm. The same procedure can also be applied to the three-dimensional Stokes problem. \par As transmission conditions for the resulting domain decomposition method of the Stokes problem we obtain natural boundary conditions. Therefore it can be implemented easily. \par A Fourier analysis and some numerical experiments show very fast convergence of the proposed algorithm. Our algorithm shows a more robust behavior than Neumann-Neumann or FETI type methods.