Algebraic conditioning analysis of the incremental unknowns preconditioner

Abstract

Abstract Incremental unknowns are efficient in the numerical solution of elliptic linear differential equations but no rigorous theoretical justification was available. Hereafter, we establish that the condition number of the incremental unknowns matrix associated to the Laplace operator is O(1/h02)O(( log h)2) where h0 is the mesh size of the coarsest grid and where h is the mesh size of the finest grid. Furthermore, if block diagonal scaling is used then the condition number of the preconditioned incremental unknowns matrix associated to the Laplace operator comes out to be O(( log h)2); last, we observe that block diagonal scaling by the Laplace operator (scaled by h02) on the coarsest grid and by 4I on the fine grids appears as an acceptable alternative.