Incremental Unknowns in finite differences in three space dimensions

The utilization of incremental unknowns was proposed as a tool for the approximation of inertial manifolds when finite differences discretizations are used. In a first step the emphasis has been put on their utilization for the solution of linear elliptic problems (more specifically in space dimension two), where they appear as a preconditioner for the corresponding linear systems. Thanks to their flexibility, they are related sometimes to the hierarchical bases in finite elements or to wavelets, and other classes of Incremental unknowns occur as well.

In this article, we describe the application of incremental unknowns for solving the Laplace problem in space dimension three. We introduce and study here the second-order incremental unknowns, and prove by deriving suitable a priori estimates that the incremental unknowns are small as expected. We then analyze the condition number of the matrix corresponding to the five-points discretization of the Laplace operator. We show that this number is $0(h^{-1}(ln h)^4)$ instead of $0(h^{-2})$ when the usual nodal unknowns are used, $h$ being the fine grid mesh size.

Min Chen (