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

TL;DR

This paper introduces a flexible and efficient computational framework within Gridap.jl for implementing various hybrid finite element methods on complex polytopal meshes, streamlining their development and application.

Contribution

It presents new abstractions for mesh representation, local assembly, and static condensation, enabling concise and efficient implementation of hybrid methods in Julia.

Findings

Framework successfully implemented for multiple PDEs

Achieves computational efficiency with Julia's JIT and lazy evaluation

Facilitates development of hybrid methods on general polytopal meshes

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.