A preconditioner with inexact element face solver for three dimensional \(p\)-version finite element methods (Q1612359)

For the approximate solution of the first boundary value problem of the Poisson equation in a three-dimensional polyhedral domain, the authors consider a \(p\)-version finite element method of tensor-product type with hexahedral elements. Subdividing the basis functions into three classes (those connected to element interiors, faces and the wirebasket) a block structure arises. The inner degrees of freedom can be eliminated locally and in parallel. For the degrees of freedom connected to element faces, \textit{L. F. Pavarino} and \textit{O. B. Widlund} [Comput. Math. Appl. 33, No.~1--2, 193--209 (1997; Zbl 0871.41020)] proposed a preconditioner containing exact solves on the element faces and needing \(O(p^4)\) operations perface. In the present work, this amount is reduced to \(O(p^3)\) using approximate face solvers based on the \(H^{1/2}\) norm of element faces -- without sacrificing the condition number of the Pavarino-Widlund preconditioner (which is \(O((1+\ln p)^2)\)).