Gradient-prolongation commutativity and graph theory

Abstract

This Note gives conditions that must be imposed to algebraic multilevel discretizations involving at the same time nodal and edge elements so that a gradient-prolongation commutativity condition will be satisfied; this condition is very important, since it characterizes the gradients of coarse nodal functions in the coarse edge function space. They will be expressed using graph theory and they provide techniques to compute approximation bases at each level.