A preconditioner with inexact element face solver for three dimensional p-version finite element methods

In this paper, we propose to modify a preconditioner developed in Pavarino and Widlund (Comput. Math. Appl. 33 (1997) 193) for three-dimensional p-version finite element or spectral element approximations by solving the sub-problems on element faces with an inexact element face solver. Such a modification reduces the cost to evaluate the action of the preconditioner from O(p4) to O(p3) per element face, where p is the polynomial degree used for approximation. Furthermore, it is shown that the coefficient matrix of the inexact solver is spectrally equivalent to the original stiffness matrix on element faces. Therefore, such a change will not affect the polylogarithmic estimate given in Pavarino and Widlund for the condition number of the preconditioned system. Numerical results are also given to confirm the theoretical analysis.