Abstract :
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)|logh|), 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.
