A Prolongation/Restriction Operator for Whitney Elements on Simplicial Meshes

The paper is mainly focused on the construction of two transfer operators between nested grids in the case of Whitney finite elements (node-, edge-, face-, or volume-based). These transfer operators, instances of what is called "chain map" in homology, have duals acting on cochains, that is to say, arrays of degrees of freedom in the context of the finite-element discretization of variational problems. We show how these duals can act as restriction/prolongation operators in a multigrid approach to such problems, especially those involving vector fields (e.g., electromagnetism). The duality between the operation of mesh refinement of a simplicial complex and that of restriction/prolongation of degrees of freedom from one mesh to a nested one is thus analyzed and explained. We use the language of p-forms, with frequent explanatory references to the more traditional vector-fields formalism.