Quadrature-Free Polytopic Discontinuous Galerkin Methods for Transport Problems

TL;DR

The paper addresses the computational bottleneck of volume integration on polytopic elements within discontinuous Galerkin methods for transport problems. It develops a quadrature-free framework based on Euler's homogeneous function theorem and Stokes' theorem, reducing volume integrals to a boundary-recursive sequence that ultimately relies on vertex data, yielding exact cubature rules. A detailed complexity analysis reveals how the costs depend on the ambient dimension $d$, the size of the polynomial space $|\mathcal{J}|$, and a facet-graph $|E(\mathcal{P})|$, with additional improvements from distance pre-computation and pruning. The approach is integrated into DGFEM discretisations of the linear transport equation and shown to outperform traditional sub-tessellation quadrature, especially for high polynomial degrees or complex (agglomerated) polytopic meshes, in both 2D and 3D tests. These results highlight a practical, scalable route to efficient transport solvers on general polytopic meshes, preserving accuracy while reducing assembly time.

Abstract

In this article we consider the application of Euler's homogeneous function theorem together with Stokes' theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two- and three-dimensions, respectively. This approach allows for the integrals to be evaluated based on only computing the values of the integrand and its derivatives at the vertices of the polytopic domain, without the need to construct a sub-tessellation of the underlying domain of interest. Here, we present a detailed analysis of the computational complexity of the proposed algorithm and show that this depends on three key factors: the ambient dimension of the underlying polytopic domain; the size of the requested polynomial space to be integrated; and the size of a directed graph related to the polytopic domain. This general approach is then employed to compute the volume integrals arising within the discontinuous Galerkin finite element approximation of the linear transport equation. Numerical experiments are presented which highlight the efficiency of the proposed algorithm when compared to standard quadrature approaches defined on a sub-tessellation of the polytopic elements.

Figures (10)

• Figure 1: Example of a tetrahedron $\mathcal{P}=T_3$ (left) and the associated recursive call graph $G(\mathcal{P})$ (right). Each vertex of $G(\mathcal{P})$ represents a facet of the tetrahedron (e.g. a vertex, an edge or a face). Edges between vertices in $G(\mathcal{P})$ denote the relationship between facets on the boundaries of other facets (e.g. the edge $ab$ lies on the boundary of the face $abc$).
• Figure 2: Effect of pruning on the recursive call graph for Algorithm \ref{['alg:monomial_integration']} in the case $\mathcal{P}=T_3$ as in Figure \ref{['fig:tetrahedron_example']}. Left-to-right: first three recursive executions of ComputeIntegrals. Blue node: facet $\mathcal{F}$ associated with current execution of $\textsc{ComputeIntegrals}(\mathcal{F},\mathcal{J})$ and choice of reference point ${\bf x}_\mathcal{F}$. Yellow nodes: facets $\mathcal{F}$ associated with previous executions of $\textsc{ComputeIntegrals}(\mathcal{F},\mathcal{J})$ and choice of reference point ${\bf x}_\mathcal{F}$. Red nodes: facets $\mathcal{F}$ eliminated from recursion as a result of pruning; i.e., unvisited facets with $\mathop{\mathrm{dist}}\limits(\mathcal{F},{\bf x}_{\mathcal{F}'})=0$ for some previously-selected reference point ${\bf x}_{\mathcal{F}'}$.
• Figure 3: Time complexities of the main loops in Algorithms \ref{['alg:stiffness_assembly_quadrature']} and \ref{['alg:stiffness_assembly_quadfree']} as a function of the degree of approximation $p$. It is assumed that $N=N_{opt}$ quadrature points are used in Algorithm \ref{['alg:stiffness_assembly_quadrature']}. Top row: number of times the operations within the main loops of Algorithms \ref{['alg:stiffness_assembly_quadrature']} and \ref{['alg:stiffness_assembly_quadfree']} are executed. Bottom row: total number of floating-point operations computed within the main loops of Algorithms \ref{['alg:stiffness_assembly_quadrature']} and \ref{['alg:stiffness_assembly_quadfree']}. Left column: $d=2$. Right column: $d=3$.
• Figure 4: CPU times taken by the quadrature-based and quadrature-free-based methods to evaluate $\mathcal{I}_{n,p}$ for $p=2,4,8,16,32$ on a regular $n$-gon. Left: $n=5$. Right: $n=16$.
• Figure 5: CPU times taken by the quadrature-based and quadrature-free-based methods to evaluate $\mathcal{I}_{n,p}$ for $5\le n\le 16$ and fixed $p$. Left: $p=4$. Right: $p=32$.

Theorems & Definitions (14)

• Remark 1: Termination of Algorithm \ref{['alg:monomial_integration']}
• Theorem 1: Time and space complexity of Algorithm \ref{['alg:monomial_integration']}
• proof
• Remark 2: Simplifications for $\dim(\mathcal{F})=d$ and $\dim(\mathcal{F})=0$
• Remark 3: Facet lattice of $\mathcal{P}$
• Remark 4: Algorithm \ref{['alg:monomial_integration']} as a depth-first search
• Theorem 2: Complexity of Algorithm 1 for $d=2,3$
• proof
• Remark 5: Extension to non-convex polyhedra
• Lemma 1: Time complexity of distance pre-computation