Differential geometry and multigrid for the div-grad, curl-curl and grad-div equations

This paper is concerned with the application of principles of differential geometry in multigrid for the div-grad, curl-curl and grad-div equations. First, the discrete counterpart of the formulas for edge, face and volume elements are used to derive a sequence of a commuting edge, face and volume prolongator from an arbitrary partition of unity nodal prolongator. The implied coarse topology and the normalization of the prolongators are analyzed, and it is proved that they form a discrete de Rham sequence if they are normalized. Numerical results are presented for the resulting edge pro-longator. It is shown that this edge prolongator is a generalization of the Reitzinger-Schöberl prolongator. Next, the partition of unity and commutation properties are used to prove that all matrices in a multigrid hierarchy for the considered equations can be factorized as a matrix product separating the metric and topological information. Finally, those properties are identified as requirements for the multigrid restriction to reflect the typical topological characteristics of the div-grad and curl-curl equations. Differential geometry and multigrid for the div-grad, curl-curl and grad-div equations