Preconditioning for Domain Decomposition through Function Approximation

A new approach was presented in [M. Mu, SIAM J. Sci. Comput., 16 (1995), to appear] for constructing preconditioners through a function approximation for the domain decomposition-based preconditioned conjugate gradient method. This work extends the approach to more general cases where grids may be nonuniform, elliptic operators may have variable coefficients (but are separable and self-adjoint), and geometric domains may be nonrectangular. The theory of expressing the Schur complement as a function of a simple interface matrix is established. The approximation to this complicated function by a simple function is discussed and the corresponding error bound is given. Preconditioning a nonrectangular domain problem is done by first reducing it to a rectangular domain problem, and then applying the theory developed here for the rectangular domain case. Accurate error bounds are given by using the results in [SIAM J. Numer. Anal., 28 (1991), pp. 378–391] for typical domains, such as L-, T-, and C-shaped ones. Numerical results are also reported to illustrate the efficiency of this approach.