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

2026-02-28

AI Summary

This work addresses the dual challenges of interface abstraction and computational efficiency in the practical implementation of nonconforming hybrid polyhedral finite element methods on general polyhedral meshes. Building upon Gridap.jl, the authors propose a unified framework featuring high-level abstractions tailored for polyhedral grids. By representing meshes as graph structures, incorporating local projection operators, employing automatic static condensation, and leveraging lazy evaluation combined with just-in-time compilation, the framework seamlessly supports a range of formulations—including hybridizable discontinuous Galerkin, high-order hybrid, and weak Galerkin methods. The approach maintains code simplicity while efficiently realizing diverse nonconforming polyhedral discretizations for Poisson’s equation, linear elasticity, incompressible Stokes flow, and optimal control problems, thereby demonstrating both generality and computational performance.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

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.

Problem

• hybrid finite element methods
• polytopal meshes
• non-conforming methods
• partial differential equations
• static condensation

Innovation

• hybrid finite element methods
• polytopal meshes
• static condensation
• Gridap.jl
• graph-based mesh representation