[Paper Review] Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes

Daniele Boffi, Daniele A. Di Pietro|arXiv (Cornell University)|Sep 15, 2016

Advanced Numerical Methods in Computational MathematicsEngineering54 references22 citations

TL;DR

This paper presents a unified formulation for mixed and primal discontinuous skeletal methods on polytopal meshes, unifying various high-order methods through globally coupled discontinuous skeletal degrees of freedom. The key contribution is establishing equivalence between mixed and primal methods under suitable conditions and delivering optimal convergence rates in both energy and $L^2$ norms via a unified analysis.

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.

Motivation & Objective

• To unify mixed and primal discontinuous skeletal methods on polytopal meshes through a common framework.
• To identify conditions under which mixed and primal methods are equivalent, particularly regarding gradient approximation.
• To provide a unified convergence analysis yielding optimal error estimates in energy and $L^2$ norms.
• To demonstrate that numerous existing methods, including hybrid high-order, virtual element, and mimetic finite difference schemes, fit within the proposed framework.

Proposed method

• The method defines two families of discretizations—mixed and primal—based on three polynomial degrees for element and skeletal degrees of freedom.
• A stabilization bilinear form is introduced, requiring only stability (uniform norm equivalence) and polynomial consistency for validity.
• The framework uses discontinuous skeletal degrees of freedom that are single-valued on faces but discontinuous at vertices/edges, enabling global coupling.
• Equivalence between mixed and primal methods is proven: every mixed method has a corresponding equivalent primal method, and vice versa if gradients are approximated in suitable spaces.
• A unified convergence analysis is conducted using energy and $L^2$ error estimates, leveraging elliptic regularity and approximation properties of projectors.
• The analysis relies on key tools: broken elliptic projections, $L^2$-projectors, and discrete integration by parts via Tonti diagrams and discrete differential operators.

Experimental results

Research questions

• RQ1 Can mixed and primal discontinuous skeletal methods be unified under a single formulation on polytopal meshes?
• RQ2 Under what conditions is a mixed method equivalent to a primal method, and vice versa?
• RQ3 What are the optimal convergence rates for the unified formulation in energy and $L^2$ norms?
• RQ4 How do existing methods such as hybrid high-order, virtual element, and mimetic finite difference schemes fit into this unified framework?
• RQ5 What are the minimal assumptions on the stabilization bilinear form to ensure stability and convergence?

Key findings

• The paper establishes that for any mixed discontinuous skeletal method, there exists a corresponding equivalent primal method, and the converse holds if gradients are approximated in suitable finite-dimensional spaces.
• Optimal convergence rates of order $h^{k+2}$ are proven in the energy norm for the primal formulation and in the $L^2$ norm for the mixed formulation, under appropriate regularity assumptions.
• The unified analysis delivers optimal error estimates for both energy and $L^2$ norms, with convergence rates dependent on the polynomial degree $k$ and the regularity of the solution.
• The stabilization bilinear form needs only to satisfy two simple conditions: stability (uniform norm equivalence) and polynomial consistency, making it easy to verify for new schemes.
• The framework encompasses a wide range of existing methods, including Raviart–Thomas and Crouzeix–Raviart elements on simplicial meshes, and extends their equivalence to polytopal meshes.
• The analysis confirms that the method achieves optimal convergence for $k imes k$ polynomial degrees in the energy norm and $k+1$ in the $L^2$ norm, with error bounds scaling as $h^{k+2}$.