A natural language framework for non-conforming hybrid polytopal methods in Gridap.jl

Abstract

Hybrid finite element methods such as hybridizable discontinuous Galerkin, hybrid high-order and weak Galerkin have emerged as powerful techniques for solving partial differential equations on general polytopal meshes. Despite their diverse mathematical origins, these methods share a common computational structure involving hybrid discrete spaces, local projection operators and static condensation. This work presents a comprehensive framework for implementing such methods within the Gridap finite element library. We introduce new abstractions for polytopal mesh representation using graph-based structures, broken polynomial spaces on arbitrary mesh entities, patch-based local assembly for cell-wise linear systems, high-level local operator construction and automated static condensation. These abstractions enable concise implementations of hybrid methods while maintaining computational efficiency through Julia's just-in-time compilation and Gridap's lazy evaluation strategies. We demonstrate the framework through implementations of several non-conforming polytopal methods for the Poisson problem, linear elasticity, incompressible Stokes flow and optimal control on polytopal meshes.