Key challenges and bridges among convergence analysis techniques for polytopal methods
For researchers in numerical analysis, this work provides a unified theoretical foundation for analyzing polytopal methods, but it is incremental as it synthesizes existing techniques rather than introducing a new paradigm.
The paper introduces a novel framework based on conforming liftings to unify convergence analysis techniques for polytopal methods, bridging virtual-element and fully discrete approaches. It demonstrates the framework on a model problem and connects to discrete differential complexes.
Polytopal methods provide a flexible framework for the numerical approximation of partial differential equations on general meshes. Their convergence analysis raises specific challenges due to their inherently non-conforming nature and, in many cases, the fully discrete nature of their solution. Two main techniques are considered: the virtual-function approach, used, e.g., in the context of Virtual Element Methods, and the fully discrete approach, which underlies, e.g., the Discrete de Rham method. We introduce here a novel framework based on the notion of conforming liftings, namely bounded and consistent mappings from the discrete space into the continuous space. This approach bridges the virtual and fully discrete viewpoints, clarifies the role of norm equivalence for virtual functions, and leads to a decomposition of the consistency error usable for polytopal methods. The three approaches are demonstrated on a model problem, which provides the opportunity to discuss relevant technical points. Bridges with the convergence properties of discrete differential complexes are also built.