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

TL;DR

This paper introduces a high-order discontinuous Galerkin method on unstructured polyhedral meshes using subgrid basis functions, enabling efficient, quadrature-free computations for complex 3D PDEs.

Contribution

It extends agglomerated finite element basis functions to polyhedral grids and integrates a local spacetime predictor with artificial viscosity for robustness.

Findings

Achieves high-order accuracy on complex polyhedral meshes.

Demonstrates robustness with shock-capturing via artificial viscosity.

Validates effectiveness on 3D Euler and Navier-Stokes benchmarks.

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.