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.

翻译：混合有限元方法，如可杂交间断伽辽金法、混合高阶法和弱伽辽金法，已成为在一般多面体网格上求解偏微分方程的有力技术。尽管这些方法数学起源各异，但它们共享一种通用的计算结构，涉及混合离散空间、局部投影算子和静力凝聚。本研究提出了在Gridap有限元库中实现此类方法的综合框架。我们引入了基于图结构的多面体网格表示新抽象、任意网格实体上的分片多项式空间、基于单元块的局部装配用于单元线性系统、高层局部算子构造以及自动化静力凝聚。这些抽象使得混合方法的实现简洁明了，同时通过Julia的即时编译和Gridap的惰性求值策略保持计算效率。我们通过在多面体网格上实现泊松问题、线弹性、不可压缩斯托克斯流及最优控制的若干非协调多面体方法，展示了该框架的有效性。