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

TL;DR

This work presents a comprehensive framework for implementing hybrid finite element methods within the Gridap finite element library, and introduces 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.

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.

Figures (8)

• Figure 1: Illustration of the local hho discrete space on a hexagonal cell $T$ for polynomial degrees $(k,l)=(0,0),(1,1),(2,2)$. The space consists of polynomials defined on the cell (grey) and polynomials defined on each facet (black).
• Figure 2: Different representations for a scutoid. Left: 3D polyhedron representation. Middle: Associated rotation system $R$. Right: Vertex adjacency list $\texttt{adj}(R)$.
• Figure 3: Generation of a 2D Voronoi mesh with Gridap. Left: Julia code to generate the Voronoi mesh. Right: Visualisation of the generated mesh. The initial simplexified Cartesian mesh is shown in light grey for reference.
• Figure 4: dg method convergence for the Poisson problem: $L^2$-error (left) and $H^1$-error (right) versus mesh size $h$ for polynomial degrees $k=1,2,3$. Reference lines show the expected convergence rates in the respective norms. Results generated using the driver from Listing \ref{['lst:example_dg_poisson']} on Voronoi meshes constructed as described in Figure \ref{['fig:voronoi']}.
• Figure 5: hdg method convergence for the Poisson problem: $L^2$ error (left) and $H^1$ error (right) versus mesh size $h$ for polynomial degrees $k=0,1,2,3$. Reference lines show the theoretical convergence rates $O(h^{k+2})$ for $L^2$ and $O(h^{k+1})$ for $H^1$. Results generated using the driver from Listing \ref{['lst:example_hdg_poisson']} on Voronoi meshes constructed as described in Figure \ref{['fig:voronoi']}.