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

TL;DR

This paper introduces a unified framework for mixed and primal discontinuous skeletal methods on polytopal meshes, establishing their equivalence and providing a comprehensive convergence analysis with optimal error estimates.

Contribution

It develops a unified formulation for mixed and primal discontinuous skeletal methods, proves their equivalence, and offers a unified convergence analysis with optimal error bounds.

Findings

Mixed and primal methods are equivalent under certain conditions.

The framework includes several existing methods as special cases.

Optimal error estimates are established in energy and L2 norms.

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 also carried out delivering optimal error estimates in both energy- and $L^2$-norms.