Continuous 3D Finite Element Subgrid Basis Functions for Discontinuous Galerkin Methods on Polyhedral Meshes

Abstract: We present a novel high-order accurate nodal discontinuous Galerkin (DG) method for solving nonlinear hyperbolic systems of partial differential equations (PDEs) on fully unstructured three-dimensional polyhedral meshes. A mesh generator is firstly discussed in detail, which ensures the generation of admissible control volumes. For the first time, we then extend the concept of agglomerated finite element (AFE) basis functions to polyhedral grids. In this context, the discrete solution is represented within each polyhedral element using piecewise continuous polynomials of degree N, defined on an internal tetrahedral subgrid. The AFE basis functions are therefore constructed by agglomerating standard finite element basis functions on each sub-tetrahedron of the computational cell. This allows for the precomputation of universal local matrices (mass and stiffness) on the reference element given by the unit tetrahedron, enabling a quadrature-free implementation that remains efficient even on highly irregular polyhedral meshes. High-order of accuracy in time is achieved using a local spacetime Galerkin predictor as part of the ADER approach, applied independently within each polyhedral element. To ensure robustness in the presence of discontinuities such as shocks, an artificial viscosity limiter is embedded into the numerical scheme, allowing for controlled dissipation and stabilization without compromising the overall accuracy in smooth regions. To demonstrate the robustness and accuracy of the method, we validate it through different three-dimensional benchmark problems for the compressible Euler and Navier-Stokes equations.

• The paper introduces a high-order ADER-AFE-DG method on arbitrary 3D polyhedral meshes that overcomes traditional meshing limitations.
• It employs a quadrature-free implementation using precomputed universal reference matrices to enhance computational efficiency.
• An effective artificial viscosity limiter stabilizes discontinuities, yielding optimal convergence and robustness in benchmark tests.

High-Order ADER-AFE-DG Methods on Polyhedral Meshes: Formulation, Implementation, and Benchmarking

Introduction and Motivation

The paper presents a high-order nodal discontinuous Galerkin (DG) method for solving nonlinear hyperbolic conservation laws, specifically the compressible Euler and Navier-Stokes equations, on fully unstructured three-dimensional polyhedral meshes (2605.00619). The significance of this work lies in its extension of the agglomerated finite element (AFE) basis functions—previously restricted to two-dimensional polygonal grids—to arbitrary three-dimensional polyhedral grids. This approach overcomes limitations inherent to regular tetrahedral or hexahedral meshing, such as geometric inflexibility, by enabling accurate and efficient simulation over complex domains. A further highlight is the quadrature-free implementation through universal reference element matrices, ensuring computational efficiency even for highly irregular polytopal grids.

Mesh Generation and Polyhedral Subgrid Construction

The core challenge in polyhedral mesh-based DG methods is the absence of a standard reference element, precluding conventional precomputation of mass and stiffness matrices. The paper addresses this with a robust two-stage mesh generation algorithm:

1. Constrained Delaunay Tetrahedralization (CDT) is used to produce an initial subdivision of the domain.
2. Local Polyhedral Replacement (LPR) systematically merges collections of incident tetrahedra into polyhedra, ensuring that each polyhedral element admits an internal tetrahedral subgrid.

Each polyhedral cell, possibly non-convex, is decomposed into a conforming set of tetrahedra serving as the support for high-order finite element basis functions. The mesh generation scheme maintains topological validity and is computationally efficient, as validated by empirical complexity estimates.

Numerical Scheme: ADER-AFE-DG Formulation

The numerical solution is expanded within each polyhedral element using nodal Lagrange polynomials defined on its internal tetrahedral subgrid, forming the Agglomerated Finite Element (AFE) basis:

• Locality: Basis functions are continuous within each polyhedral cell but discontinuous across element boundaries, which ensures interface localization of numerical fluxes as required in DG.
• Universal Matrix Precomputation: Because all subgrid elements are mapped from a universal reference tetrahedron, mass and stiffness matrices (and consequently all quadrature) can be precomputed, yielding a quadrature-free implementation.

Time integration leverages the ADER (Arbitrary high-order DERivatives) approach, combining a local space-time Galerkin predictor (evolving the polynomial solution inside each cell) with a one-step explicit corrector and robust Rusanov-type fluxes. The time step restriction follows the standard DG Courant–Friedrichs–Lewy (CFL) stability limit with explicit dependence on polynomial degree and mesh geometry.

Artificial Viscosity Limiter

High-order DG methods are susceptible to non-physical oscillations near discontinuities due to the Gibbs phenomenon. To address this, the method employs a selective artificial viscosity strategy:

• A shock sensor, based on the local divergence of the velocity field and minimum speed of sound, flags "troubled" cells.
• Only in these cells, an artificial viscosity is added to reach a unit mesh Reynolds number, stabilizing the solution locally without global accuracy loss.

This construct is fully geometric-aware, naturally integrating the complex face and edge structure of three-dimensional polyhedral meshes.

Numerical Benchmarks and Results

The paper details a suite of stringent benchmark problems showcasing the method’s accuracy, robustness, and efficiency:

• Stationary and Travelling Vortex Problems: On unstructured polyhedral meshes, the method demonstrates optimal convergence rates (up to spatial order $N+1$ for polynomial degree $N=3$) in both $L_1$ and $L_2$ norms, confirming high-order spatial and space-time accuracy.
• First Stokes Problem: Accurate results for various viscosities and mesh refinements verify correct handling of viscous terms and dissipative effects.
• Three-Dimensional Spherical Explosion: The method robustly captures spherical shock propagation, with artificial viscosity effectively suppressing spurious oscillations near discontinuities. Radial symmetry and expected wave locations/strengths are well-matched with reference solutions.
• Taylor-Green Vortex: The method recovers known analytical results for velocity and vorticity, maintaining optimal convergence and resolving complex vortical structures over time.

In all tests, the observed convergence rates match the theoretically predicted orders. The method remains robust in the presence of shocks and steep gradients without excessive numerical dissipation.

Implications and Future Perspectives

The proposed high-order ADER-AFE-DG framework introduces practical advantages for computational fluid dynamics and simulation of complex physical systems:

• Computational Efficiency: The quadrature-free approach dramatically reduces run-time costs, especially relevant for high-order, large-scale 3D computations.
• Geometric Flexibility: Adoption of arbitrary polyhedral meshes allows for effective geometry fitting in real-world domains, including locally refined/adaptive regions.
• Parallelization Potential: The structure is conducive to GPU acceleration and distributed-memory parallelism, further enhancing its suitability for exascale simulations.
• Application Scope: The approach can be directly extended to Arbitrary-Lagrangian-Eulerian (ALE) frameworks, solid mechanics, and multi-material phenomena.

Potential future directions include incorporation of more advanced monolithic limiting strategies (such as subcell convex property preserving fluxes), further improving the balance of accuracy and robustness in strong shock regimes, as well as systematic extension to fluid-structure interaction and multiphysics problems.

Conclusion

The paper establishes a theoretically sound and practically efficient high-order discontinuous Galerkin methodology on unstructured polyhedral meshes for three-dimensional nonlinear hyperbolic systems. By extending the AFE paradigm to 3D, introducing a quadrature-free ADER-DG assembly, and implementing an effective artificial viscosity limiter, the method offers a flexible, accurate, and robust solution framework. It paves the way for high-fidelity simulation in domains with complex geometries, advancing the state of numerical PDE integration for real-world engineering and scientific applications (2605.00619).