A natural language framework for non-conforming hybrid polytopal methods in Gridap.jl
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911475678838784 |
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| author | Manyer, Jordi Tushar, Jai Badia, Santiago |
| author_facet | Manyer, Jordi Tushar, Jai Badia, Santiago |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00880 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A natural language framework for non-conforming hybrid polytopal methods in Gridap.jl Manyer, Jordi Tushar, Jai Badia, Santiago Mathematical Software Numerical Analysis 68-04 (Primary) 65Y15, 65N30 (Secondary) G.4; G.1.8 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. |
| title | A natural language framework for non-conforming hybrid polytopal methods in Gridap.jl |
| topic | Mathematical Software Numerical Analysis 68-04 (Primary) 65Y15, 65N30 (Secondary) G.4; G.1.8 |
| url | https://arxiv.org/abs/2603.00880 |