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

The main obstacle for obtaining fast domain decomposition solvers for spectral element discretizations of second order elliptic equations was the lack of fast solvers for local internal problems on subdomains of decomposition and their faces. As recently shown by Korneev and Rytov, such solvers can be derived on the basis of a specific interrelation between stiffness matrices of the spectral and hierarchical p reference elements (coordinate polynomials of the latter are tensor products of the integrated Legendre’s polynomials). This interrelation allows us to develop fast solvers for discretizations by spectral elements, which are quite similar in basic features to those developed for discretizations by hierarchical elements. Using these facts and preceding findings on the wire basket preconditioners, we present a domain decomposition preconditioner-solver for spectral element discretizations of second order elliptic equations in 3-d domains which is almost optimal in the total arithmetical cost.