Discontinuous Skeletal Gradient Discretisation Methods on polytopal meshes
Provides a unified framework for several existing methods (Hybrid High-Order, Mimetic Finite Difference, Virtual Element) on polytopal meshes, but the contribution is incremental as it extends known techniques.
This work develops arbitrary-order Discontinuous Skeletal Gradient Discretisations on general polytopal meshes, proving convergence for elliptic and parabolic problems. Numerical examples validate the approach.
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.