Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes^∗,∗∗

¹ Universitàdegli Studi di Pavia, Dipartimento di Matematica “Felice Casorati”, 27100 Pavia, Italy.
daniele.boffi@unipv.it
² Institut Montpelliérain Alexander Grothendieck, CNRS, Univ. Montpellier, France.
daniele.di-pietro@umontpellier.fr

Received: 13 September 2016
Revised: 19 June 2017
Accepted: 19 July 2017

Abstract

We propose in this work a unified formulation of mixed and primal discretization methods on polyhedral meshes hinging on globally coupled degrees of freedom that are discontinuous polynomials on the mesh skeleton. To emphasize this feature, these methods are referred to here as discontinuous skeletal. As a starting point, we define two families of discretizations corresponding, respectively, to mixed and primal formulations of discontinuous skeletal methods. Each family is uniquely identified by prescribing three polynomial degrees defining the degrees of freedom, and a stabilization bilinear form which has to satisfy two properties of simple verification: stability and polynomial consistency. Several examples of methods available in the recent literature are shown to belong to either one of those families. We then prove new equivalence results that build a bridge between the two families of methods. Precisely, we show that for any mixed method there exists a corresponding equivalent primal method, and the converse is true provided that the gradients are approximated in suitable spaces. A unified convergence analysis is carried out delivering optimal error estimates in both energy- and L²-norms.