An Explicit Harmonic Extension for The Constant-like Basis And Its Application to Domain Decompositions1

In this paper we are concerned with substructuring methods for the second-order elliptic problems in three-dimensional domains. We first design a simple and completely explicit nearly harmonic extension for the constant-like basis function (i.e., the face basis function), and then define a coarse subspace based on this nearly harmonic extension. Own to the resulting coarse solver, we develop a kind of substructuring preconditioner with inexact solvers. We show that the condition number of the preconditioned system grows only as the logarithm of the dimension of the local problem associated with an individual substructure, and is independent of possible jumps of the coefficient in the elliptic equation. Numerical experiments confirms the theoretical results