Fast domain decomposition algorithm for discretizations of 3-d elliptic equations by spectral elements

The main obstacle for obtaining fast domain decomposition solvers for the spectral element discretizations of the 2-nd order elliptic equations was the lack of fast solvers for local internal problems on subdomains of decomposition and their faces. As was recently shown by Korneev/Rytov, such solvers can be derived on the basis of the specific interrelation between the stiffness matrices of the spectral and hierarchical p reference elements. The coordinate polynomials of the latter are tensor products of the integrated Legendre’s polynomials. This interrelation allows to apply to the spectral element discretizations fast solvers which in some basic features are quite similar to those developed for the discretizations by the hierarchical elements. Using these facts and the preceding results on the wire basket preconditioners, we present an almost optimal in total arithmetical cost domain decomposition preconditioner-solver for the spectral element discretizations of the 2-nd order elliptic equations in 3-d domains. 2000 Mathematics Subject Classification: 65N22; 65M30; 65N55.