Incremental unknowns and graph techniques in three space dimensions

For three space dimensions, from now on, we show that the condition number of the incremental unknowns matrix associated to the Laplace operator is O(1/H2)O((1/h)|log h|), where H 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, the condition number of the preconditioned incremental unknowns matrix associated to the Laplace operator turns out to be O(1/h), these conditioning results proved by using a generic algebraic conditioning analysis-based upon graph techniques--and by setting up a discrete inequality. In contrast, the condition number of the nodal unknowns matrix associated to the Laplace operator is O (1/h2). Therefore, the incremental unknowns preconditioner is an efficient preconditioner in three space dimensions.