Discontinuous Skeletal Gradient Discretisation Methods on polytopal meshes

Daniele Di Pietro (1) , Jérôme Droniou (2) , Gianmarco Manzini (3)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 School of Mathematical Sciences [Clayton]
3 LANL - Los Alamos National Laboratory

In this work we develop arbitrary-order Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomial on the mesh skeleton. The key ingredient is a high-order gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high-order residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.

Mots clés: Gradient discretisation methods; gradient schemes; high-order Mimetic Finite Difference methods; Hybrid High-Order methods; Virtual Element methods; non-linear problems

Domaines: Analyse numérique [math.NA]

Daniele Di Pietro, Jérôme Droniou, Gianmarco Manzini. Discontinuous Skeletal Gradient Discretisation Methods on polytopal meshes. 2017. ⟨hal-01564598v1⟩