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Journal of Scientific Computing

, Volume 77, Issue 3, pp 1339–1370 | Cite as

Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods

  • Paola F. Antonietti
  • Paul Houston
  • Giorgio Pennesi
Open Access
Article
  • 144 Downloads

Abstract

In this paper we present efficient quadrature rules for the numerical approximation of integrals of polynomial functions over general polygonal/polyhedral elements that do not require an explicit construction of a sub-tessellation into triangular/tetrahedral elements. The method is based on successive application of Stokes’ theorem; thereby, the underlying integral may be evaluated using only the values of the integrand and its derivatives at the vertices of the polytopic domain, and hence leads to an exact cubature rule whose quadrature points are the vertices of the polytope. We demonstrate the capabilities of the proposed approach by efficiently computing the stiffness and mass matrices arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations.

Keywords

Numerical integration Polygonal/polyhedral meshes hp-discontinuous Galerkin method 

Mathematics Subject Classification

65D30 65D32 65L60 

1 Introduction

In recent years the exploitation of computational meshes composed of polygonal and polyhedral elements has become very popular in the field of numerical methods for partial differential equations. Indeed, the flexibility offered by polygonal/polyhedral elements allows for the design of efficient computational grids when the underlying problem is characterized by a strong complexity of the physical domain, such as, for example, in geophysical applications, fluid-structure interaction, or crack propagation problems. Moreover, the possibility to adopt computational meshes with hanging nodes is included in this framework by observing that, for example, a classical quadrilateral element with a hanging node on one of its edges can be treated as a pentagon with two aligned edges. Several conforming numerical discretization methods which admit polygonal/polyhedral meshes have been proposed within the current literature; here, we mention, for example, the Composite Finite Element Method [5, 41, 42], the Mimetic Finite Difference (MFD) method [4, 18, 19, 20, 21, 44], the Polygonal Finite Element Method [63], the Extended Finite Element Method [37, 64], the Virtual Element Method (VEM) [10, 11, 15, 16, 17] and the Hybrid High-Order (HHO) method [33, 34, 35]. In the non-conforming setting, we mention Discontinuous Galerkin (DG) methods [1, 2, 3, 6, 9, 14, 22, 23, 24, 25, 25], Hybridizable DG methods [29, 30, 31, 32], non-conforming VEM [8, 13, 26], and the Gradient Schemes [36]; here the possibility of defining local polynomial discrete spaces follows naturally with the flexibility provided by polytopic meshes.

One of the key aspects concerning the development of efficient finite element discretizations with polygonal/polyhedral grids is the construction of quadrature formulae for the approximate computation of the terms appearing in the underlying weak formulation. Indeed, the design of efficient quadrature rules for the numerical computation of integrals over general shaped polytopes is far from being a trivial task. The classical and most widely employed technique for the integration over polytopes is the Sub-Tessellation method, cf. [38, 51, 62]; here, the domain of integration is subdivided into standard-shaped elements, such as triangular/quadrilateral elements in 2D or tetrahedral/hexahedral elements in 3D, whereby standard efficient quadrature rules are employed, cf. [50, 60, 70], and also [71] and [48], for an interpolation technique based on the same idea. On the one hand this technique is easy to implement, however, it is generally computationally expensive, particularly for high order polynomials, since the number of function evaluations may be very large.

For this reason, the development of quadrature rules that avoid sub-tessellation is an active research field. Several approaches have been proposed; in particular, we mention [43, 54, 67, 68], for example. One interesting method in this direction is represented by the Moment Fitting Equation technique, firstly proposed by Lyness and Monegato in [49], for the construction of quadrature rules on polygons featuring the same symmetry as the regular hexagon. Generalizations to convex and non-convex polygons and polyhedra were proposed by Mousavi et al. in [53]. Here, starting from an initial quadrature rule, given, for example, by the sub-tessellation method described above, an iterative node elimination algorithm is performed based on employing the least-squares Newton method [69] in order to minimise the number of quadrature points while retaining exact integration. Further improvements of the moment fitting equation algorithm can also be found in [52] and [61]. While this method is optimal with respect to the number of function evaluations, the nodes and weights must be stored for every polygon, thus affecting memory efficiency. An alternative approach designed to overcome the limitations of the sub-tessellation approach is based on employing the generalized version of Stokes’ theorem; here, the exploitation of Stokes’ theorem reduces the integral over a polytope to an integration over its boundary; see [66] for details. For the two-dimensional case, in [59], Sommariva and Vianello proposed a quadrature rule based on employing Green’s theorem. In particular, if an x- or y-primitive of the integrand is available (as for bivariate polynomial functions), the integral over the polygon is reduced to a sum of line integrations over its edges. When the primitive is not known, this method does not directly require a sub-tessellation of the polygon, but a careful choice of the parameters in the proposed formula leads to a cubature rule that can be viewed as a particular sub-tessellation of the polygon itself. However, it is not possible to guarantee that all of the quadrature points lie inside the domain of integration. An alternative and very efficient formula has been proposed by Lasserre in [47] for the integration of homogeneous functions over convex polytopes. This technique has been recently extended to general convex and non-convex polytopes in [27]. The essential idea here is to exploit the generalized Stokes’ theorem together with Euler’s homogeneous function theorem, cf. [58], in order to reduce the integration over a polytope only to boundary evaluations. The main difference with respect to the work presented in [59] is the possibility to apply the same idea recursively, leading to a quadrature formula which exactly evaluates integrals over a polygon/polyhedron by employing only point-evaluations of the integrand and its derivatives at the vertices of the polytope.

In this article we extend the approach of [27] to the efficient computation of the volume/face integral terms appearing in the discrete weak formulation of second-order elliptic problems, discretized by means of high-order DG methods. We point out that our approach is completely general and can be directly applied to other discretization schemes, such as VEM, HHO, Hybridisable DG, and MFD, for example. We focus on the DG approach presented in [25], where the local polynomial discrete spaces are defined based on employing the bounded box technique [39]. We show that our integration approach leads to a considerable improvement in the performance compared to classical quadrature algorithms based on sub-tessellation, in both two- and three-dimensions. The outline of this article is as follows: in Sect. 2 we recall the work introduced in [27], and outline how this approach can be utilized to efficiently compute the integral of d-variate polynomial functions over general polytopes. In Sect. 3 we introduce the interior penalty DG formulation for the numerical approximation of a second-order diffusion–reaction equation on general polytopic meshes. In Sect. 4 we outline the exploitation of the method presented in Sect. 2 for the assembly of the mass and stiffness matrices appearing in the DG formulation. Several two- and three-dimensional numerical results are presented in Sect. 5 in order to show the efficiency of the proposed approach. Finally, in Sect. 6 we summarise the work undertaken in this article and discuss future extensions.

2 Integrating Polynomials over General Polygons/Polyhedra

In this section we review the procedure introduced by Chin et al. in [27] for the integration of homogeneous functions over a polytopic domain. To this end, we consider the numerical computation of \( \int _{{\mathscr {P}}} g({\mathbf {x}}) \mathrm {d} {\mathbf {x}},\) where
  • \({\mathscr {P}} \subset \mathbb {R}^d \textit{, }d=2,3\), is a closed polytope, whose boundary \(\partial {\mathscr {P}} \) is defined by m\((d-1)\)-dimensional faces\({\mathcal {F}}_i,\ i=1,\ldots ,m\). Each face\({\mathcal {F}}_i\) lies in a hyperplane \({\mathcal {H}}_i\) identified by a vector \({\mathbf {a}}_i \in \mathbb {R}^d \) and a scalar number \(b_i\), such that
    $$\begin{aligned} {\mathbf {x}} \in {\mathcal {H}}_i \iff {\mathbf {a}}_i \cdot {\mathbf {x}} = b_i, \quad i=1,\ldots ,m. \end{aligned}$$
    (1)
    We observe that \({\mathbf {a}}_i\), \(i=1,\ldots ,m\), can be chosen as the unit outward normal vector to \({\mathcal {F}}_i\), \(i=1,\ldots ,m\), respectively, relative to \({\mathscr {P}} \), cf. Figs. 1 and 2.
  • \(g:{\mathscr {P}} \rightarrow \mathbb {R}\) is a homogeneous function of degree \(q \in \mathbb {R}\), i.e., for all \(\lambda > 0\), \(g(\lambda {\mathbf {x}}) = \lambda ^q g({\mathbf {x}})\) for all \({\mathbf {x}} \in {\mathscr {P}} \).

We recall that Euler’s homogeneous function theorem [58] states that, if g is a homogeneous function of degree \(q \ge 0\), then the following identity holds:
$$\begin{aligned} q\ g({\mathbf {x}}) = \nabla g({\mathbf {x}}) \cdot {\mathbf {x}} \quad \forall {\mathbf {x}} \in {\mathscr {P}}. \end{aligned}$$
(2)
Next we introduce the generalized Stokes’ theorem, which can be stated as follows, cf. [66]: given a generic vector field \({\mathbf {X}}: {\mathscr {P}} \rightarrow \mathbb {R}^d \), the following identity holds
$$\begin{aligned} \int _{{\mathscr {P}}} (\nabla \cdot {\mathbf {X}}({\mathbf {x}})) g({\mathbf {x}}) \mathrm {d} {\mathbf {x}} + \int _{{\mathscr {P}}} \nabla g({\mathbf {x}}) \cdot {\mathbf {X}}({\mathbf {x}}) \mathrm {d} {\mathbf {x}} = \int _{\partial {\mathscr {P}}} {\mathbf {X}}({\mathbf {x}}) \cdot {\mathbf {n}}({\mathbf {x}}) g({\mathbf {x}}) \mathrm {d} \sigma , \end{aligned}$$
(3)
where \({\mathbf {n}}\) is the unit outward normal vector to \({\mathscr {P}} \) and \(\mathrm {d} \sigma \) denotes the \((d-1)\)-dimensional (surface) measure. Selecting \({\mathbf {X}} = {\mathbf {x}} \) in (3), and employing (2), gives
$$\begin{aligned} \int _{{\mathscr {P}}} g({\mathbf {x}}) \mathrm {d} {\mathbf {x}} = \frac{1}{d+q} \int _{\partial {\mathscr {P}}} {\mathbf {x}} \cdot {\mathbf {n}}({\mathbf {x}}) g({\mathbf {x}}) \mathrm {d} \sigma = \frac{1}{d+q} \sum _{i=1}^m b_i \int _{{\mathcal {F}}_i} g({\mathbf {x}})\ \mathrm {d} \sigma . \end{aligned}$$
(4)
Fig. 1

Example of a two-dimensional polytope \({\mathscr {P}} \) and its face Open image in new window . The hyperplane Open image in new window is defined by the local origin \({\mathbf {x}} _{0,i}\) and the vector \({\mathbf {e}}_{i1}\)

Fig. 2

The dodecahedron \({\mathscr {P}} \) with pentagonal faces and the face Open image in new window with unit outward normal vector \({\mathbf {n}}_i\). Here, Open image in new window has five edges Open image in new window , and five unit outward normal vectors \({\mathbf {n}}_{ij},\ j=1,\ldots ,5\), lying on the plane Open image in new window . The hyperplane Open image in new window is identified by the local origin \({\mathbf {x}} _{0,i}\) and the orthonormal vectors \({\mathbf {e}}_{i1}, {\mathbf {e}}_{i2}\)

Equation (4) states that if g is homogeneous, then the integral of g over a polytope \({\mathscr {P}} \) can be evaluated by computing the integral of the same function over the boundary faces Open image in new window , \(i=1,\ldots ,m\). By applying Stokes’ theorem recursively, we can further reduce each term Open image in new window , to the integration over \(\partial {\mathcal {F}}_i,\ i=1,\ldots ,m\), respectively. To this end, Stokes’ theorem needs to be applied on the hyperplane \({\mathcal {H}}_i,\ i=1,\ldots ,m\), in which each \({\mathcal {F}}_i,\ i=1,\ldots ,m\), lies, respectively. In order to proceed, let \(\varvec{\gamma } : \mathbb {R}^{d-1} \rightarrow \mathbb {R}^d \) be the function which expresses a generic point \(\tilde{{\mathbf {x}}} = (\tilde{x}_1,\ldots ,\tilde{x}_{d-1})^\top \in \mathbb {R}^{d-1}\) as a point in \(\mathbb {R}^d \) that lies on \({\mathcal {H}}_i,\ i=1,\ldots ,m\), i.e.,
$$\begin{aligned} \tilde{{\mathbf {x}}} \longmapsto \varvec{\gamma }(\tilde{{\mathbf {x}}}) = {\mathbf {x}} _{0,i} + \sum _{n=1}^{d-1} \tilde{x} _n {\mathbf {e}}_{in},\quad \text {with }\ {\mathbf {e}}_{in} \in \mathbb {R}^d,\ \ {\mathbf {e}}_{in} \cdot {\mathbf {e}}_{im} = \delta _{nm}. \end{aligned}$$
Here, \({\mathbf {x}} _{0,i} \in {\mathcal {H}}_i,\ i=1,\ldots ,m\), is an arbitrary point which represents the origin of the coordinate system on \({\mathcal {H}}_i\), and \(\{ {\mathbf {e}}_{in} \}_{n=1}^{d-1}\) is an orthonormal basis on \({\mathcal {H}}_i,\ i=1,\ldots ,m\); see Figs. 1 and 2 for two- and three-dimensional examples, respectively. Notice that \({\mathbf {x}} _{0,i}\) does not have to lie inside Open image in new window . Let Open image in new window such that Open image in new window , then the following identity holds:Before outlining the details regarding the recursive application of the Stokes’ Theorem to (4), we first require the following lemma.

Lemma 1

Let Open image in new window be the vertices/edges of Open image in new window for \(d=2,3\), respectively, and let \({\mathbf {n}}_{ij}\) be the unit outward normal vectors to Open image in new window lying in Open image in new window . Moreover, let Open image in new window be the preimage of Open image in new window with respect to the map \(\varvec{\gamma }\), and \(\widetilde{{\mathbf {n}}}_{ij}\) be the corresponding unit outward normal vector. Then, the following holds
$$\begin{aligned} \tilde{{\mathbf {n}}}_{ij} = \underline{{\mathbf {E}}}^\top {\mathbf {n}}_{ij}, \quad i=1,\ldots ,m,\ j=1,\ldots ,m_i, \end{aligned}$$
where \(\underline{{\mathbf {E}}} \in \mathbb {R}^{d \times (d-1)}\), whose columns are the vectors \(\{ {\mathbf {e}}_{in} \}_{n=1}^{d-1},\ i=1,\ldots ,m\).

Proof

We first note that employing the definition of \(\varvec{\gamma }\) we have that
$$\begin{aligned} {\mathbf {x}} = \varvec{\gamma }(\tilde{{\mathbf {x}}}) = {\mathbf {x}} _{0,i} + \underline{{\mathbf {E}}} \tilde{{\mathbf {x}}}, \quad \text{ i.e., } \quad {\mathbf {x}}- {\mathbf {x}} _{0,i} = \underline{{\mathbf {E}}} \tilde{{\mathbf {x}}}. \end{aligned}$$
(6)
The proof now follows immediately from simple linear algebra considerations; for full details, we refer to [7]. \(\square \)

Given identity (5) and Lemma 1, we can prove the following result.

Proposition 1

Let Open image in new window be a face of the polytope \({\mathscr {P}} \), and let Open image in new window , \(j=1,\ldots ,m_i\), be the planar/straight faces/edges such that Open image in new window for some \(m_i \in \mathbb {N}\). Then, for any homogeneous function g, of degree \(q \ge 0\), the following identity holdswhere \(d_{ij}\) denotes the Euclidean distance between Open image in new window and \({\mathbf {x}} _{0,i}\), Open image in new window , is arbitrary, \(i=1,\ldots ,m\), and \(\mathrm {d} \nu \) denotes the \((d-2)\)-dimensional (surface) measure.

Proof

If we denote by \(\nabla _i = \bigl [\frac{\partial }{\partial \tilde{x} _1}, \ldots , \frac{\partial }{\partial \tilde{x} _{d-1}} \bigr ]^\top \) the gradient operator on the hyperplane \({\widetilde{\mathcal {H}}_i},\ i=1,\ldots ,m,\) with respect to the coordinate system \((\tilde{x} _1,\ldots ,\tilde{x} _{d-1})\), then, upon application of Stokes’ theorem, we havewhere \(\tilde{{\mathbf {n}}}\) is the unit outward normal vector of \(\widetilde{{\mathcal {F}}_i}\) and \(\widetilde{{\mathbf {X}}}\) is a vector field on \(\mathbb {R}^{d-1}\). Next, we transform (7) back to the original coordinate system. To this end, denoting \(\underline{{\mathbf {E}}} \in \mathbb {R}^{d \times (d-1)}\) to be the matrix whose columns are the vectors \(\{ {\mathbf {e}}_{in} \}_{n=1}^{d-1}\), we observe that, if we choose \(\widetilde{{\mathbf {X}}} = \tilde{{\mathbf {x}}} \), then its divergence is \(\nabla _i \cdot \widetilde{{\mathbf {X}}} = d-1\). Exploiting (6), the term \(\nabla _i g(\varvec{\gamma }(\tilde{{\mathbf {x}}}))\) can be written as follows:
$$\begin{aligned} \nabla _i g(\varvec{\gamma }(\tilde{{\mathbf {x}}})) = \begin{bmatrix} \frac{\partial \gamma _1}{\partial \tilde{x} _1}&\frac{\partial \gamma _2}{\partial \tilde{x} _1}&\cdots&\frac{\partial \gamma _d}{\partial \tilde{x} _1} \\ \frac{\partial \gamma _1}{\partial \tilde{x} _2}&\frac{\partial \gamma _2}{\partial \tilde{x} _2}&\cdots&\frac{\partial \gamma _d}{\partial \tilde{x} _2} \\ \vdots&\vdots&\ddots&\vdots \\ \frac{\partial \gamma _1}{\partial \tilde{x} _{d-1}}&\frac{\partial \gamma _2}{\partial \tilde{x} _{d-1}}&\cdots&\frac{\partial \gamma _d}{\partial \tilde{x} _{d-1}} \end{bmatrix} \left[ \begin{array}{c} \frac{\partial g}{\partial \tilde{x} _1} \\ \frac{\partial g}{\partial \tilde{x} _2} \\ \vdots \\ \frac{\partial g}{\partial \tilde{x} _d} \end{array} \right] = (\underline{{\mathbf {E}}}^\top \nabla g)(\varvec{\gamma }(\tilde{{\mathbf {x}}})). \end{aligned}$$
(8)
Exploiting (6) and (8), we can write Open image in new window and Open image in new window asrespectively. Employing Lemma 1, together with (6), we have thatWe observe that the term \(({\mathbf {x}}- {\mathbf {x}} _{0,i}) \cdot {\mathbf {n}}_{ij} \) is constant for any Open image in new window , and that it represents the Euclidean distance between Open image in new window and \({\mathbf {x}} _{0,i}\); thereby, we define \(d_{ij} = ({\mathbf {x}}- {\mathbf {x}} _{0,i}) \cdot {\mathbf {n}}_{ij}\). From the above identities (9), (10) and (11) we deduce the statement of the Proposition. \(\square \)
Using Proposition 1, together with Eq. (4), we obtain the following identitywhere we recall that Open image in new window and Open image in new window , for \(i=1,\ldots ,m\).

Remark 1

If \(d=2\), then Open image in new window is a point and (12) states that the integral of g on \({\mathscr {P}} \) can be computed by vertex-evaluations of the integrand plus a line integration of the partial derivative of g. If \(d=3\) we can apply Stokes’ Theorem recursively to Open image in new window . Proceeding as before, we getwhere Open image in new window , \({\mathbf {x}} _{0,ij}\) is an arbitrarily chosen origin for Open image in new window , and \(d_{ijk}\) is the Euclidean distance between Open image in new window and \({\mathbf {x}} _{0,ij}\).
Table 1

Polytopic domains of integration Open image in new window considered in Algorithm 1 as a function of the dimension d

 

\(N = 3\)

\(N=2\)

\(N=1\)

\(N=0\)

\(d=3\)

Open image in new window is a polyhedron

Open image in new window is a polygon

Open image in new window is an edge

Open image in new window is a point

\(d=2\)

 

Open image in new window is a polygon

Open image in new window is an edge

Open image in new window is a point

\(d=1\)

  

Open image in new window is an interval

Open image in new window is a point

In view of the application of Proposition 1 to finite element methods, we are interested in the integration of a particular class of homogeneous functions, namely polynomial homogeneous functions of the form
$$\begin{aligned} g({\mathbf {x}}) = x_1^{k_1} x_2^{k_2} \ldots x_d^{k_d}, \quad \text {where } k_n \in \mathbb {N}_0 \text { for }n=1,\ldots ,d. \end{aligned}$$
In this case, g is a homogeneous function of degree \(q=k_1 +\cdots + k_d\), and the general partial derivative \(\frac{\partial g}{\partial x_n}\) is a homogeneous function of degree \(q-1\). With this in mind, it is possible to recursively apply formula (12) to the terms involving the integration of the derivatives of g. To this end, we write Open image in new window , be a N-polytopic domain of integration, with \(N=1,\ldots ,d\), and let Open image in new window , where each Open image in new window is a \((N-1)\)-polytopic domain. When \(N=d\) and \(d=2,3\), Open image in new window will be an edge or a face, respectively; see Table 1 for details. We define the functionwhich returns the integral of the polynomial \(x_1^{k_1}\ldots x_d^{k_d}\) over Open image in new window , where \(\mathrm {d} \sigma _N\) is the N-dimensional (surface) measure, \(N=1,2,\ldots ,d\). According to Proposition 1, the recursive definition of the function Open image in new window is given in Algorithm 1.

Remark 2

With a slight abuse of notation, when \(1 \le N \le d-1\), in Algorithm 1 (and for the purposes of the following discussion), the point \({\mathbf {x}} _{0} = (x_{0,1},\ldots ,x_{0,d})^\top \) denotes an arbitrarily chosen origin for the coordinate system which defines the N-polytope Open image in new window and \(d_i\) represents the Euclidean distance between the \((N-1)\)-polytopes Open image in new window , which form the boundary of Open image in new window , and \({\mathbf {x}} _{0},\ i=1,\ldots ,m\). Furthermore, in Algorithm 1, \(b_i,\ i=1,\ldots ,m,\) is the same constant appearing in (1). Here it can be evaluated as \(b_i = {\mathbf {n}}_i \cdot {\mathbf {v}},\) where \({\mathbf {v}}\) is a vertex of Open image in new window and \({\mathbf {n}}_i\) is the unit outward normal vector, \(i=1,\ldots ,m\).

Remark 3

We point out that in (12), cf. also (13), the shape of the underlying polytope can be general: indeed, nonconvex simply-connected domains Open image in new window are admissable.

Fig. 3

Triangle \(({\mathscr {P}} _1)\)

Fig. 4

Irregular polygon with 5 faces \(({\mathscr {P}} _2)\)

Fig. 5

Irregular polygon with 15 faces \(({\mathscr {P}} _3)\)

2.1 Integration of Bivariate Polynomials over Polygonal Domains

In order to test the performance of the method proposed in Algorithm 1, we consider the integration of bivariate homogeneous functions on a given polygon \({\mathscr {P}} \subset \mathbb {R}^2\) based on using the three different approaches:
A.1

Recursive algorithm described in Sect. 2, based on the formula (13): Open image in new window cf. Algorithm 1.

A.2

Use of the formula (4) together with numerical integration employed for the evaluation of the edge integrals with known one-dimensional Gaussian quadrature rules, as recently proposed in [28];

A.3

Sub-tessellation technique: the domain of integration \({\mathscr {P}} \) is firstly decomposed into triangles where standard efficient quadrature rules are then employed.

Table 2

Coordinates of the polygons of Figs. 34, and 5

 

Vertex

x-cordinates

y-cordinates

\({\mathscr {P}} _1\)

1

\(-\) 1.000000000000000

\(-\) 1.000000000000000

2

1.000000000000000

0.000000000000000

3

\(-\) 1.000000000000000

1.000000000000000

\({\mathscr {P}} _2\)

1

\(-\) 0.666666666666667

\(-\) 0.789473684210526

2

0.555555555555556

\(-\) 1.000000000000000

3

1.000000000000000

\(-\) 0.052631578947368

4

\(-\) 0.555555555555556

1.000000000000000

5

\(-\) 1.000000000000000

\(-\) 0.157894736842105

\({\mathscr {P}} _3\)

1

0.413048522141662

0.781696234443715

2

0.024879797655533

0.415324992429711

3

\(-\) 0.082799691823524

0.688810136531751

4

\(-\) 0.533191422779328

1.000000000000000

5

\(-\) 0.553573605852999

0.580958514816226

6

\(-\) 0.972432940212767

0.734117068746903

7

\(-\) 1.000000000000000

0.238078507228890

8

\(-\) 0.789986179147920

0.012425068086110

9

\(-\) 0.627452906935866

\(-\) 0.636532897516109

10

\(-\) 0.452662174765764

\(-\) 1.000000000000000

11

\(-\) 0.069106265580153

\(-\) 0.289054989277619

12

0.141448047807069

\(-\) 0.464417038155806

13

1.000000000000000

\(-\) 0.245698820584615

14

0.363704451489016

\(-\) 0.134079689960635

15

0.627086024018283

\(-\) 0.110940423607648

Table 3

The approximated values of the integral over the three polygons in Figs. 34 and 5 obtained with approach A.1

 

\({\mathscr {P}} _1\)

\({\mathscr {P}} _2\)

\({\mathscr {P}} _3\)

Open image in new window

0

\(-\,0.0020324991\)

\(-\,0.002589861\)

Open image in new window

0.0111339078

\(7.4274779926\times 10^{-5}\)

\(1.5738050178\times 10^{-4}\)

Open image in new window

0.0030396808

\(6.0738145408\times 10^{-8}\)

\(1.3793481020\times 10^{-6}\)

Open image in new window

\(7.9534562047 \times 10^{-4}\)

\(2.2238524572\times 10^{-12}\)

\(4.2588831784\times 10^{-10}\)

Open image in new window

0

\(-\,2.0911953867\times 10^{-4}\)

0.0014996521

Open image in new window

0

\(-\,1.3797380205\times 10^{-5}\)

\(7.0356275077\times 10^{-4}\)

Open image in new window

0

\(-\,7.9203571311\times 10^{-7}\)

\(2.5065856538\times 10^{-4}\)

Open image in new window

\(-\,0.005890191\)

\(8.08469022058\times 10^{-5}\)

\(-\,1.330384913\times 10^{-4}\)

Open image in new window

\(-\,0.001868889\)

\(4.37593748009\times 10^{-5}\)

\(-\,3.963064075\times 10^{-5}\)

Table 4

CPU times as a function of the integrand and the integration domain \({\mathscr {P}} \) for the three approaches A.1, A.2 and A.3

 

\({\mathscr {P}} _1\)

\({\mathscr {P}} _2\)

\({\mathscr {P}} _3\)

 

A.1

A.2

A.3

A.1

A.2

A.3

A.1

A.2

A.3

\(x^5y^5\)

0.054

0.159

0.616

0.083

0.244

0.973

0.227

0.678

2.856

\(x^{10}y^{10}\)

0.078

0.221

1.359

0.123

0.328

2.321

0.351

0.939

7.301

\(x^{20}y^{20}\)

0.124

0.344

4.060

0.207

0.540

7.399

0.580

1.498

22.70

\(x^{40}y^{40}\)

0.208

0.578

14.79

0.377

0.934

27.24

1.073

2.671

86.63

\(x^{10}y^5\)

0.064

0.191

0.999

0.081

0.296

1.699

0.237

0.833

5.125

\(x^{20}y^5\)

0.078

0.240

1.955

0.089

0.412

3.690

0.274

1.093

10.99

\(x^{40}y^5\)

0.107

0.363

4.975

0.085

0.616

9.504

0.332

1.680

29.40

\(x^5y^{20}\)

0.052

0.244

1.971

0.085

0.412

3.662

0.243

1.117

11.07

\(x^5y^{40}\)

0.051

0.365

5.009

0.082

0.597

9.295

0.272

1.673

29.17

We test the three different approaches for integrating bivariate polynomials of different polynomial degrees on the triangle depicted in Fig. 3 and the two irregular polygons shown in Figs. 4 and 5, cf. Table 2 for the list of coordinates for each domain; the actual values of the integrals are given in Table 3. In Table 4 we show the average CPU-time taken to evaluate the underlying integral using each method. We point out that, for each integrand and each integration domain \({\mathscr {P}} \), the relative errors between the output of the three different approaches are of the order of machine precision; that is, all three algorithms return the exact integral up to roundoff error. For completeness, we note that the times for A.1 include the computation of \(b_i,\ {\mathbf {n}}_i\), and \(d_{ij}\), \(j=1,\ldots ,m_i\), \(i=1,\ldots ,m\). For A.2 we take into account the evaluation of \(b_i,\ {\mathbf {n}}_i\), \(i=1,\ldots ,m\), and the one-time computation of the one-dimensional quadrature defined on \((-1,1)\), consisting of Open image in new window nodes and weights, employed for the line integrations. Here, we select Open image in new window , in order to guarantee the exact integration of \(x^k y^l\). The times for A.3 include the one-time computation of the Open image in new window nodes and weights on the reference triangle, where Open image in new window is selected as in A.2, the time required for sub-tessellation, as well as the time needed for numerical integration on each sub-triangle. The results shown in Table 4 illustrate that the sub-tessellation approach A.3 is the slowest while the proposed method A.1 is the fastest for all of the considered cases; in particular, we highlight that, even for just a single domain of integration, the former method is between one- to two-orders of magnitude slower than the latter approach proposed in this article. Moreover, when the integration domain consists of a triangle, our algorithm A.1 still outperforms classical quadrature rules, cf. A.3, even though in this case no sub-tessellation is undertaken. When comparing A.1 and A.2, we observe that the former algorithm is again superior in terms of CPU time in comparison with the latter approach; this difference seems to grow when the exponents k and l of the integrand function \(x^ky^l\) are very different. This is because in A.1 we have made an optimal selection of the points \({\mathbf {x}} _{0,i} = (x_{0i,1},x_{0i,2})^\top \), \(i=1,\ldots ,m\), appearing in (12). Indeed, performing the geometric reduction of the edges of the domain of integration, we then choose \(x_{0i,1} = 0\) or \(x_{0i,2}=0\), \(i=1,\ldots ,m\), if the exponents of the integrand function \(x^ky^l\) are \(k \ge l\) or \(k<l\), respectively. The choice \(x_{0i,1} = 0\) or \(x_{0i,2}=0\), \(i=1,\ldots ,m\), allows us to avoid the recursive calls to the function Open image in new window related to the x- or y-partial derivatives, respectively. In this way the approach A.1 is able to exploit the form of the integrand in order to optimize the evaluation of the corresponding integral. To explore this issue further, in the following section we consider the computational complexity of A.1 in both the cases when an optimal and non-optimal selection of the points \({\mathbf {x}} _{0,i}\), \(i=1,\ldots ,m\), is made.

2.2 Computational Complexity of Algorithm 1

The computational complexity of Algorithm 1, which is employed in A.1, depends in general on the number of recursive calls of the function Open image in new window . In particular, using the short-hand notation introduced in Remark 2, the selection of the points \({\mathbf {x}} _{0} = (x_{0,1},\ldots ,x_{0,d})^\top \), which are used to define the origin of the coordinate system of each N-polytope Open image in new window which defines the facets of \({\mathscr {P}} \) is crucial. In general, any \((d-1)\)-dimensional hyperplane in \(\mathbb {R}^d\) possesses a non-empty intersection with some axis of the Cartesian reference system, which means that it is always possible to choose \((d-1)\) components of \({\mathbf {x}} _{0}\) as zero. Without loss of generality we select \(x_{0,r} = \nicefrac {b_i}{n_{i,r}}\) and \(x_{0,s} = 0\) for \(s \ne r\), where \(b_i\) and \({\mathbf {n}}_i\) are as defined in Remark 2, and \(r \in \{ 1,\ldots ,d\}\) is chosen so that \( k_r = \min \{ k_1,\ldots ,k_d\}.\)

Remark 4

In general, if Open image in new window is a N-polytopic domain in \(\mathbb {R}^d\), then at most N components of Open image in new window can be selected to be zero.

In this way, the selection of \(k_r\) essentially fixes the number of recursive calls of Open image in new window in Algorithm 1. More precisley, we write Open image in new window to denote the number of FLOPs to perform Open image in new window , and let \(C_N\) be the number of FLOPs required by Open image in new window , without considering the recursive calls of Open image in new window to itself. With this in mind, let us consider the following two examples:
  • Set \(d=2\) and assume \(k_1 \le k_2\), so that we can choose \(x_{0,1}\ne 0\) and \(x_{0,2} = 0\) on each of the edges of \({\mathscr {P}} \). Then, according to Algorithm 1 we have and where we have denoted the vertices of the edge Open image in new window as \({\mathbf {v}}_{i1}\) and \({\mathbf {v}}_{i2}\). Hence, In general, for \(d=2\) we deduce that
  • Set \(d=3\) and assume \(k_1 = \min \{k_1,k_2,k_3\}\), so that we may select \(x_{0,1} \ne 0\) and \(x_{0,2} = x_{0,3} = 0\) on each of the faces of \({\mathscr {P}} \). Thereby, employing Algorithm 1 we deduce that where, for each \(i=1,\ldots ,m\), Here, the computational complexity of Open image in new window depends on the choice of \({\mathbf {x}} _{0} \equiv {\mathbf {x}} _{0,ij}\) which defines the origin of the coordinate system for Open image in new window , \(j=1,\ldots ,m_i\), \(i=1,\ldots ,m\). According to Remark 4, two components of \({\mathbf {x}} _{0,ij}\) can possibly be different from zero, which implies that the complexity of Algorithm 1 increases exponentially when \(d=3\). However, it is possible to modify Algorithm 1 in order to avoid the double recursive calls which cause this exponential complexity. In particular, in Sect. 2.3 we propose an alternative algorithm which exploits the same idea of Algorithm 1 and allows us to overcome this issue.
Fig. 6

Comparison of the number of FLOPs required to evaluate \(\int _{{\mathscr {P}}} x^{k_1} y^{k_2} \mathrm {d} \mathbf{x}\), based on fixing \(k_1\) and varying \(k_2 \in \{0,\ldots ,50\}\): a Quadrature free method A.1; b Sub-tessellation method A.3

Fig. 7

Comparison of the number of FLOPs required to evaluate \(\int _{{\mathscr {P}}} x^{k} y^{k} \mathrm {d} \mathbf{x}\) as k increases employing both the quadrature free and sub-tessellation methods: a Excludes function evaluations; b Total cost including function evaluations

In order to confirm (14), we use the tool [56] to measure the number of FLOPs required to exactly compute \(\int _{{\mathscr {P}}} x^{k_1} y^{k_2} \mathrm {d} \mathbf{x}\); moreover, comparisons will also be made with A.3. To simplify the presentation, the polygon \({\mathscr {P}} \) is selected to be the triangle with vertices \((-1,0.3)\), \((1,-1)\), and (0.3, 1); thereby, A.3 does not require the computation of a sub-tessellation. In Fig. 6, we plot the number of FLOPs needed to evaluate \(\int _{{\mathscr {P}}} x^{k_1} y^{k_2}\mathrm {d} \mathbf{x}\) by fixing \(k_1\) and varying \(k_2 \in \{0,\ldots ,50\}\) employing both A.1 and A.3. In particular, Fig. 6a shows that the number of FLOPs required by the quadrature free method A.1 growths linearly with respect to \(k_2\) when \(k_1>k_2\) and becomes constant as \(k_2\) increases when \(k_1 \le k_2\). Figure 7 confirms the asymptotic behaviour of the two algorithms in the case when \(k_1=k_2\); here, the number of FLOPs required by the sub-tessellation method is reported in both the case when the cost of the evaluation of the quadrature nodes and weights employing the function gauleg, cf. [55], for example, is included/excluded. In particular, we show results both in the case when the cost of the function evaluations is excluded, cf. Fig. 7a, as well as the total number of FLOPs required by each algorithm to exactly evaluate \(\int _{{\mathscr {P}}} x^{k} y^{k}\mathrm {d} \mathbf{x}\), cf. Fig. 7b. As expected, the computational complexity of A.3 grows as \({\mathcal O}(k^2)\) and \({\mathcal O}(k^3)\) in these two latter cases, respectively, while the cost of the quadrature free method is always \({\mathcal O}(k)\) as k increases.

Thus far, the numerical results presented for the proposed quadrature free method have assumed that the points \({\mathbf {x}} _{0}\), which are used to define the origin of the coordinate system of each N-polytope Open image in new window which defines the facets of \({\mathscr {P}} \), has been chosen in an optimal manner to ensure that the number of recursive calls of Open image in new window , cf. Algorithm 1, is minimized. Indeed, a sub-optimal choice of these points leads to an exponential growth in the number of recursive calls of the function Open image in new window in Algorithm 1. For example, if \(d=2\) the non-optimal choice of \({\mathbf {x}} _{0}\) implies that each call of Open image in new window with \(N = 1\) leads to a double recursive call of Open image in new window , up to when a zero exponent \(k_1\) or \(k_2\) appears as input. In particular, if \(k_1 = k_2 = k\), it is possible to show that the number of FLOPs required by the quadrature free method grows as \({\mathcal O}(2^{2k-1})\), as k increases, cf. Fig. 8. In the following section, we present an alternative implementation of the quadrature free algorithm which avoids this exponential growth, irrespective of the selection of the points \({\mathbf {x}} _0\).
Fig. 8

Number of FLOPs required to evaluate \(\int _{{\mathscr {P}}} x^{k} y^{k} \mathrm {d} \mathbf{x}\) as k increases, based on employing the quadrature free method, with a sub-optimal choice of \({\mathbf {x}} _{0}\)

Fig. 9

Number of FLOPs required to evaluate \(\{ \int _\kappa x^{k_1} y^{k_2} \mathrm {d} {\mathbf {x}} ~ \forall \ k_1,k_2 \ge 0, ~ k_1+k_2 \le p \}\) based on employing Algorithm 1 (with an optimal selection of the points \({\mathbf {x}} _0\)), and Algorithm 2

2.3 Integration of Families of Monomial Functions

In the context of employing the quadrature free approach within a finite element method, in practice we are not interested in integrating a single monomial function, but instead an entire family of monomials, which, for example, form a basis for the space of polynomials of a given degree over a given polytopic element \(\kappa \) which belongs to the underlying computational mesh. For example, when \(d=2\), let us consider the evaluation of
$$\begin{aligned} \int _{\kappa } x^{k_1} y^{k_2} \mathrm {d} \mathbf{x} \quad \forall \ k_1,k_2 \ge 0, \quad k_1+k_2 \le p. \end{aligned}$$
(15)
We note that even when employing the Approach A.1 with an optimal choice of the points \({\mathbf {x}} _{0}\), the total number of FLOPs required for the computation of (15) is approximately Open image in new window , as p increases.
To improve the dependence on p we propose an alternative approach, cf. Algorithm 2; this is based on the observation that, using the notation of Algorithm 1, if the values of Open image in new window , \(j=1,\ldots ,m\), Open image in new window , for \(1 \le N \le d-1\), in Algorithm 1, have already been computed, then the computation of Open image in new window is extremely cheap. Indeed, since we must store the integrals of all the monomials on \(\kappa \) anyway, we can start by computing and storing \(\int _{\kappa } x^{k_1} y^{k_2} \mathrm {d} x_1 \mathrm {d} x_2\) related to the lower degrees \(k_1,k_2\) and \(N=1\), then exploit these values in order to compute the integrals with higher degrees \(k_1,k_2\) and higher dimension N of the integration domain Open image in new window . This leads to an algorithm, whereby the number of FLOPs required to compute and store \(\{ \int _{\kappa } x_1^{k_1}\ldots x_d^{k_d}\mathrm {d} \sigma _d(x_1,\ldots ,x_d),\ k_1,\ldots ,k_d \ge 0, ~k_1+k_2+\cdots +k_d \le p \}\) is of order Open image in new window , as p increases, irrespective of the selection of choice of the points \({\mathbf {x}} _{0}\). In Fig. 9 we now compare these two approaches for \(d=2\), when the underlying element is selected to be the triangular region employed in the previous section. Here, we compare Algorithm 1, with an optimal selection of the points \({\mathbf {x}} _{0}\), with Algorithm 2, where in the latter case the points \({\mathbf {x}} _{0}\) are simply selected to be equal to the first vertex defining each edge; here, we clearly observe the predicted increase in FLOPs of Open image in new window and Open image in new window , as p increases, for each of the two algorithms, respectively.

3 Application to hp-Version DG Methods

We consider the following elliptic model problem, given by: find u such that
$$\begin{aligned} \begin{aligned} -\Delta u + u&=f \, \text {in }\varOmega \;,\\ u&=0 \, \text {on }\partial \varOmega \;, \end{aligned} \end{aligned}$$
(16)
where \(\varOmega \subset \mathbb {R}^d \), \(d = 2,3\), is a polygonal/polyhedral domain with boundary \(\partial \varOmega \) and f is a given function in \(L^2(\varOmega )\).

In order to discretize problem (16), we introduce a partition Open image in new window of the domain \(\varOmega \), which consists of disjoint (possibly non-convex) open polygonal/polyhedral elements \(\kappa \) of diameter \(h_{\kappa }\), such that Open image in new window . We denote the mesh size of Open image in new window by Open image in new window . Furthermore, we define the faces of the mesh Open image in new window as the planar/straight intersections of the \((d-1)\)-dimensional facets of neighbouring elements. This implies that, for \(d=2\), a face consists of a line segment, while for \(d=3\), the faces of Open image in new window are general shaped polygons; without loss of generality, for the definition of the proceeding DG method we assume that the faces are \((d-1)\)-dimensional simplices, cf. [24, 25] for a discussion of this issue. In order to introduce the DG formulation, it is helpful to distinguish between boundary and interior element faces, denoted by \({\mathcal {F}}_h^B\) and \({\mathcal {F}}_h^I\), respectively. In particular, we observe that \(F \subset \partial \varOmega \) for \(F \in {\mathcal {F}}_h^B\), while for any \(F \in {\mathcal {F}}_h^I\) we assume that \(F \subset \partial \kappa ^{\pm }\), where \(\kappa ^{\pm }\) are two adjacent elements in Open image in new window . Furthermore, we write \({\mathcal {F}}_h = {\mathcal {F}}_h^I \cup {\mathcal {F}}_h^B\) to denote the set of all mesh faces of Open image in new window . For simplicity of presentation we assume that each element Open image in new window possesses a uniformly bounded number of faces under mesh refinement, cf. [24, 25].

We associate to Open image in new window the corresponding discontinuous finite element space \(V_h\), defined by Open image in new window where Open image in new window denotes the space of polynomials of total degree at most \(p_{\kappa } \ge 1\) on Open image in new window , cf. [24, 25].

In order to define the DG method, we introduce the jump and average operators:where \(v^\pm \) and \(\varvec{\tau }^\pm \) denote the traces of sufficiently smooth scalar- and vector-valued functions v and \(\varvec{\tau }\), respectively, on F taken from the interior of \(\kappa ^\pm \), respectively, and \({\mathbf {n}}^\pm \) are the unit outward normal vectors to \(\partial \kappa ^\pm \), respectively, cf. [12].
We then consider the bilinear form Open image in new window , corresponding to the symmetric interior penalty DG method, defined bywhere \(\nabla _h\) denotes the broken gradient operator, defined elementwise, and Open image in new window denotes the interior penalty stabilization function, whose precise definition, based on the analysis introduced in [24, 25], is given below. To this end, we first need the following definition.

Definition 1

Let Open image in new window be the subset of elements Open image in new window such that each Open image in new window can be covered by at most Open image in new window shape-regular simplices Open image in new window , such thatfor all Open image in new window , for some Open image in new window , where \(C_{as}\) and \(c_{as}\) are positive constants, independent of \(\kappa \) and Open image in new window .

Given Definition 1, we recall the following inverse inequality, cf. [24, 25].

Lemma 2

Let Open image in new window , \(F\subset \partial \kappa \) denote one of its faces, and Open image in new window be defined as in Definition 1. Then, for each Open image in new window , we have the inverse estimate
$$\begin{aligned} \Vert v \Vert _{L^2(F)}^2 \le C_{INV}(p_{\kappa },\kappa ,F) \frac{p_{\kappa }^2 |F|}{|\kappa |} \Vert v \Vert _{L^2(\kappa )}^2, \end{aligned}$$
whereand \(\kappa _{\flat }^F\) denotes a d-dimensional simplex contained in \(\kappa \) which shares the face F with Open image in new window . Furthermore, \(C_{inv}\) is a positive constant, which if Open image in new window depends on the shape regularity of the covering of \(\kappa \) given in Definition 1, but is always independent of \(| \kappa | / \sup _{\kappa _{\flat }^F \subset \kappa } | \kappa _{\flat }^F |,\ p_{\kappa }\) and v.

Based on Lemma 2, together with the analysis presented in [24, 25], the parameter \(\alpha _h\) can be defined as follows.

Definition 2

Let Open image in new window be defined facewise bywith \(C_{\alpha }>C_{\alpha }^{min}\), where \(C_{\alpha }^{min}\) is a sufficiently large lower bound.
The DG discretization of the problem (16) is given by: find \(u_h \in V_h\) such thatBy fixing a basis \(\{ \phi _i\}_{i=1}^{N_h}\), \(N_h\) denoting the dimension of the discrete space \(V_h\), (19) can be rewritten as: find \({\mathbf {U}} \in \mathbb {R}^{N_h}\) such that
$$\begin{aligned} ({\mathbf {A}} + {\mathbf {M}} ){\mathbf {U}} = {\mathbf {f}}, \end{aligned}$$
(20)
where \({\mathbf {f}}_i = \int _{\varOmega } f\phi _i \mathrm {d} {\mathbf {x}}\ \forall i=1,\ldots ,N_h\), \({\mathbf {A}}\) is the stiffness matrix, given by Open image in new window , \({\mathbf {M}}\) is the mass matrix, and \({\mathbf {U}}\) contains the expansion coefficients of \(u_h \in V_h\) with respect to the chosen basis. In order to assemble \(({\mathbf {A}} + {\mathbf {M}} )\) we need to compute the following matrices:
$$\begin{aligned} {\mathbf {M}}_{i,j}&= \int _{\varOmega } \phi _{i} \phi _{j} \mathrm {d} {\mathbf {x}}, \quad {\mathbf {V}}_{i,j} = \int _{\varOmega } \nabla \phi _i \cdot \nabla \phi _j\ \mathrm {d} {\mathbf {x}}, \end{aligned}$$
(21)
for \(i,j=1,\ldots ,N_h\), where as before \(N_h\) denotes the dimension of the DG space \(V_h\). In particular, the stiffness matrix related to the DG approximation of problem (19) is defined as \({\mathbf {A}} = {\mathbf {V}} - {\mathbf {I}}^\top - {\mathbf {I}} + {\mathbf {S}}\).

4 Elemental Stiffness and Mass Matrices

In this section, we outline the application of Algorithm 2 for the efficient computation of the mass and stiffness matrices appearing in (20).

4.1 Shape Functions for the Discrete Space \(V_h\)

To construct the discrete space \(V_h\) we exploit the approach presented in [25], based on employing polynomial spaces defined over the bounding box of each element. More precisely, given an element Open image in new window , we first construct the Cartesian bounding box \(B_{\kappa }\), such that \(\overline{\kappa } \subset \overline{B_{\kappa }}\). Given Open image in new window , it is easy to define a linear map between \(B_{\kappa }\) and the reference element \(\hat{B} = (-1,1)^d\) as follow: \({\mathbf {F}}_{\kappa }: \hat{B} \rightarrow B_{\kappa } \text { such that } {\mathbf {F}}_{\kappa }: \hat{{\mathbf {x}}} \in \hat{B} \longmapsto {\mathbf {F}}_{\kappa }(\hat{{\mathbf {x}}}) = {\mathbf {J}}_{\kappa } \hat{{\mathbf {x}}} + {\mathbf {t}}_{\kappa },\) where \( {\mathbf {J}}_{\kappa } \in \mathbb {R}^{d \times d}\) is the Jacobi matrix of the transformation which describes the stretching in each direction, and \({\mathbf {t}}_{\kappa } \in \mathbb {R}^d \) is the translation between the point \({\mathbf {0}} \in \hat{B}\) and the baricenter of the bounded box \(B_{\kappa }\), see Fig. 10. We note that since \({\mathbf {F}}_{\kappa }\) affinely maps one bounding box to another (without rotation), the Jacobi matrix \({\mathbf {J}}_{\kappa }\) is diagonal.

Employing the map Open image in new window , we may define a standard polynomial space Open image in new window on \(B_{\kappa }\) spanned by a set of basis functions \(\{ \phi _{i,\kappa } \}\) for Open image in new window . More precisely, we denote by Open image in new window the family of one-dimensional and \(L^2\)-orthonormal Legendre polynomials, defined over \(L^2(-1,1)\), i.e.,cf. [40, 57]. We then define the basis functions for the polynomial space Open image in new window as follows: writing \(I = (i_1,i_2,\ldots ,i_d)\) to denote the multi-index used to identify each basis function \(\{ \hat{\phi }_I \}_{0 \le | I | \le p}\), where \(| I | = i_1 + \cdots + i_d,\) we have thatThen, the basis functions for the polynomial space Open image in new window are defined by using the map \({\mathbf {F}}_{\kappa }\), namely:
$$\begin{aligned} \phi _{I,\kappa }({\mathbf {x}}) = \hat{\phi }_I({\mathbf {F}}_{\kappa }^{-1}({\mathbf {x}})) \quad \forall {\mathbf {x}} \in \kappa \subset B_{\kappa } \quad \forall I: 0 \le | I | \le p_{\kappa }. \end{aligned}$$
(23)
The set Open image in new window forms a basis for the space \(V_h\). On each element Open image in new window we introduce a bijective relation between the set of multi-indices \(\{ I = (i_1,\ldots ,i_d): 0 \le | I | \le p_{\kappa } \}\) and the set \(\{ 1, 2, \ldots , N_{p_{\kappa }} \}\).

4.2 Volume Integrals Over Polytopic Mesh Elements

In the following we describe the application of Algorithm 2 to compute the entries in the local mass and element-based stiffness matrices
$$\begin{aligned} {\mathbf {M}}^{\kappa }_{i,j} = \int _{\varOmega } \phi _{i,\kappa } \phi _{j,\kappa } \mathrm {d} {\mathbf {x}},\quad {\mathbf {V}}^{\kappa }_{i,j} = \int _{\varOmega } \nabla \phi _{i,\kappa } \cdot \nabla \phi _{j,\kappa } \mathrm {d} {\mathbf {x}} \quad i,j=1,\ldots ,N_{p_{\kappa }}, \end{aligned}$$
(24)
respectively, for all Open image in new window . For simplicity of presentation, we restrict ourselves to two-dimensions, though we emphasize that the three-dimensional case is analogous, cf. Sect. 5.2 below. Since the basis functions are supported only on one element, employing the transformation \({\mathbf {F}}_{\kappa }\), we have
$$\begin{aligned} {\mathbf {M}}^{\kappa }_{i,j} = \int _{\kappa } \phi _{i,\kappa }(x,y) \phi _{j,\kappa }(x,y) \mathrm {d} {\mathbf {x}} = \int _{\hat{\kappa }} \hat{\phi }_i(\hat{x},\hat{y}) \hat{\phi }_j(\hat{x},\hat{y}) |{\mathbf {J}}_{\kappa }| \mathrm {d} \hat{{\mathbf {x}}},\ i,j=1,\ldots ,N_{p_{\kappa }}, \end{aligned}$$
where in the last integral \(\hat{\kappa } = {\mathbf {F}}_{\kappa }^{-1}(\kappa ) \subset \hat{B}\), see Fig. 10. Here, the Jacobian of the transformation \({\mathbf {F}}_{\kappa }\) is given by \(|{\mathbf {J}}_{\kappa }| = ({\mathbf {J}}_{\kappa })_{1,1} ({\mathbf {J}}_{\kappa })_{2,2}\), which is constant, due to the definition of the map. In order to employ the homogeneous function integration method described in the previous section, we need to identify the coefficients of the homogeneous polynomial expansion for the function \(\hat{\phi }_i(\hat{x},\hat{y}) \hat{\phi }_j(\hat{x},\hat{y})\). We observe that Open image in new window , and each one–dimensional Legendre polynomial can be expanded as
Fig. 10

Example of a polygonal element Open image in new window , the relative bounded box \(B_{\kappa }\), the map \({\mathbf {F}}_{\kappa }\) and \(\hat{\kappa } = {\mathbf {F}}_{\kappa }^{-1}(\kappa )\)

Therefore, we have
$$\begin{aligned} {\mathbf {M}}^{\kappa }_{i,j}&= \int _{\hat{\kappa }} \left( \sum _{m=0}^{i_1} C_{i_1,m} \hat{x} ^m\right) \left( \sum _{n=0}^{i_2} C_{i_2,n} \hat{y} ^n \right) \left( \sum _{s=0}^{j_1} C_{j_1,s} \hat{x} ^s \right) \left( \sum _{r=0}^{j_2} C_{j_2,r} \hat{y} ^r \right) |{\mathbf {J}}_{\kappa }| \mathrm {d} \hat{{\mathbf {x}}} \\&= \int _{\hat{\kappa }} \left( \sum _{k=0}^{i_1+j_1} {\mathcal {C}}_{i_1,i_2,k} \hat{x} ^k \right) \left( \sum _{l=0}^{i_2 + j_2} {\mathcal {C}}_{i_2,j_2,l} \hat{y} ^l \right) |{\mathbf {J}}_{\kappa }| \mathrm {d} \hat{{\mathbf {x}}} \\&= \sum _{k=0}^{i_1+j_1} \sum _{l=0}^{i_2 + j_2} {\mathcal {C}}_{i_1,j_1,k}\ {\mathcal {C}}_{i_2,j_2,l}\ |{\mathbf {J}}_{\kappa }| \int _{\hat{\kappa }} \hat{x} ^k \hat{y} ^l \mathrm {d} \hat{{\mathbf {x}}}. \end{aligned}$$
Here, we have written
$$\begin{aligned} {\mathcal {C}}_{i,j,k} = \sum _{n+m=k} \bigl ( C_{i,n}\ C_{j,m} \bigr ), \qquad \text {for } 0 \le i,j \le p_{\kappa }, \quad 0 \le k \le i + j. \end{aligned}$$
(26)
Notice that the coefficients \({\mathcal {C}}_{i,j,k}\) can be evaluated, once and for all, independently of the polygonal element \(\kappa \). We now consider the general element of the volume matrix \({\mathbf {V}}_{i,j}\), cf. (24). Proceeding as before, let \(I,\ J\) be the two multi-indices corresponding respectively to i and j, we haveProceeding as before, we apply a change of variables to the terms Open image in new window and Open image in new window with respect to the map \({\mathbf {F}}_{\kappa }\); thereby, we obtainFrom the definition of \({\mathbf {F}}_{\kappa }\), the inverse map is given by \({\mathbf {F}}^{-1}_{\kappa }({\mathbf {x}}) = {\mathbf {J}}_{\kappa }^{-1} ({\mathbf {x}}- {\mathbf {t}}_{\kappa })\). Then, using the definition (23) of the basis functions, we have the following characterization of the partial derivatives appearing in the terms Open image in new window and Open image in new window :
$$\begin{aligned} \frac{\partial }{\partial x} \phi _{I,\kappa }({\mathbf {x}}) = \frac{\partial \hat{\phi }_I}{\partial \hat{x}}({\mathbf {F}}^{-1}_{\kappa }({\mathbf {x}}))\ ({\mathbf {J}}^{-1}_{\kappa })_{1,1}, \quad \frac{\partial }{\partial y} \phi _{I,\kappa }({\mathbf {x}}) = \frac{\partial \hat{\phi }_I}{\partial \hat{y}}({\mathbf {F}}^{-1}_{\kappa }({\mathbf {x}}))\ ({\mathbf {J}}^{-1}_{\kappa })_{2,2} \end{aligned}$$
where we have used that \(({\mathbf {J}}^{-1}_{\kappa })_{2,1} = ({\mathbf {J}}^{-1}_{\kappa })_{1,2} = 0\) since \({\mathbf {J}}_{\kappa }\) is diagonal. Then, Open image in new window can be written as:Since \(({\mathbf {J}}^{-1}_{\kappa })_{1,1}^2 | {\mathbf {J}}_{\kappa } |\) is constant, the integrand function of term Open image in new window is a polynomial. Thereby, we have the following relation:From the expansion (25) of the Legendre polynomials, we note thatwhere the indices \(C'_{i,m}=(m+1)C_{i,m+1}\) are the coefficients for the expansion of Open image in new window . We deduce that Open image in new window if \(i_1 = 0\) or \(j_1 = 0\), andwhere \({\mathcal {C}}_{i_2,j_2,l}\) is defined in (26), and
$$\begin{aligned} {\mathcal {C}}'_{i,j,k} = \sum _{n + m = k} C'_{i,n}\ C'_{j,m}, \quad 1 \le i,j \le p_{\kappa },\ \text {for}\ 0 \le k \le i+j-2, \end{aligned}$$
with \(C'_{i,n}=(n+1)C_{i,n+1}\), \(C'_{j,m}=(m+1)C_{j,m+1}\), cf. (28), is the expansion of the derivatives of the Legendre polynomials which is computable independently of the element \(\kappa \), Open image in new window . Analogously, we deduce the following expression for the second term of Eq. (27):

4.3 Interface Integrals over Polytopic Mesh Elements

With regards the interface integrals appearing in Eq. (18), we describe the method by expanding the jump and average operators and computing each term separately, working, for simplicity, again in two-dimensions. Firstly, we discuss how to transform the integral over a physical face \(F \subset \partial \kappa \) to the corresponding integral over the face \(\hat{F} = {\mathbf {F}}_{\kappa }^{-1}(F) \subset \partial \hat{\kappa }\) on the reference rectangular element \(\hat{\kappa }\). To this end, let \(F \subset \partial \kappa \) be a face of the polygon \(\kappa \), Open image in new window , and let \({\mathbf {x}} _1 = (x_1,y_1)\) and \({\mathbf {x}} _2=(x_2,y_2)\) denote the vertices of the face, based on counter clock-wise ordering of the polygon vertices. The face \(\hat{F} = {\mathbf {F}}_{\kappa }^{-1}(F)\) is identified by the two vertices \(\hat{{\mathbf {x}}} _1 = {\mathbf {F}}_{\kappa }^{-1}({\mathbf {x}})\) and \(\hat{{\mathbf {x}}} _2 = {\mathbf {F}}_{\kappa }^{-1}({\mathbf {x}} _2).\) For a general integrable function \(g:\kappa \rightarrow \mathbb {R}\) we have
$$\begin{aligned} \int _{F} g(x,y) \mathrm {d} \sigma (x,y) = \int _{\hat{F}} g({\mathbf {F}}_{\kappa }(\hat{x},\hat{y}))\ \mathrm {d} \sigma ({\mathbf {F}}_{\kappa }(\hat{x},\hat{y})), \end{aligned}$$
where Open image in new window and Open image in new window is defined as Open image in new window where Open image in new window is the unit outward normal vector to \(\hat{F}\).
We next describe how to compute the interface integrals. From the definition of the jump and average operators, cf. (17), on each edge Open image in new window shared by the elements \(\kappa ^{\pm }\) we need to assemble
$$\begin{aligned} {\mathbf {S}}_{i,j}^{+/+}&= \int _{F} \alpha _h\ \phi _{i,\kappa ^+}\ \phi _{j,\kappa ^+}\ \mathrm {d} \sigma , \quad&{\mathbf {I}}_{i,j}^{+/+}&= \frac{1}{2} \int _{F} (\nabla \phi _{i,\kappa ^+} \cdot {\mathbf {n}}^+)\ \phi _{j,\kappa ^+}\ \mathrm {d} \sigma , \\ {\mathbf {S}}_{i,j}^{-/-}&= \int _{F} \alpha _h\ \phi _{i,\kappa ^-}\ \phi _{j,\kappa ^-}\ \mathrm {d} \sigma , \quad&{\mathbf {I}}_{i,j}^{-/-}&= \frac{1}{2} \int _{F} (\nabla \phi _{i,\kappa ^-} \cdot {\mathbf {n}}^-)\ \phi _{j,\kappa ^-}\ \mathrm {d} \sigma , \\ {\mathbf {S}}_{i,j}^{+/-}&= -\int _{F} \alpha _h\ \phi _{i,\kappa ^+}\ \phi _{j,\kappa ^-}\ \mathrm {d} \sigma , \quad&{\mathbf {I}}_{i,j}^{+/-}&= -\frac{1}{2} \int _{F} (\nabla \phi _{i,\kappa ^+} \cdot {\mathbf {n}}^+)\ \phi _{j,\kappa ^-}\ \mathrm {d} \sigma , \\ {\mathbf {S}}_{i,j}^{-/+}&= -\int _{F} \alpha _h\ \phi _{i,\kappa ^-}\ \phi _{j,\kappa ^+}\ \mathrm {d} \sigma , \quad&{\mathbf {I}}_{i,j}^{-/+}&= -\frac{1}{2} \int _{F} (\nabla \phi _{i,\kappa ^-} \cdot {\mathbf {n}}^-)\ \phi _{j,\kappa ^+}\ \mathrm {d} \sigma , \end{aligned}$$
for \(i,j=1,\ldots ,N_{p_{\kappa ^{\pm }}}\). Analogously, on the boundary face Open image in new window belonging to Open image in new window we only have to compute
$$\begin{aligned} {\mathbf {S}}_{i,j}^{+/+} = \int _{F} \alpha _h\ \phi _{i,\kappa ^+}\ \phi _{j,\kappa ^+}\ \mathrm {d} \sigma , \qquad {\mathbf {I}}_{i,j}^{+/+} = \int _{F} (\nabla \phi _{i,\kappa ^+} \cdot {\mathbf {n}}^+)\ \phi _{j,\kappa ^+}\ \mathrm {d} \sigma , \end{aligned}$$
for \(i,j=1,\ldots ,N_{p_{\kappa ^+}}\). We next show how to efficiently compute a term of the form
$$\begin{aligned} {\mathbf {S}}_{i,j}^{+/+} = \int _{F} \alpha _h\ \phi _{I,\kappa ^+}(x,y)\ \phi _{J,\kappa ^+}(x,y) \mathrm {d} \sigma , \end{aligned}$$
where IJ are the suitable multi-indices associated to \(i,j=1,\ldots ,N_{p_{\kappa ^+}}\), respectively. Proceeding as before, we haveAnalogously, we haveFor the term Open image in new window , we directly apply the definition of the basis function, and obtainwhile for the term Open image in new window we haveIn order to obtain a homogeneous polynomial expansion for Open image in new window we have to write explicitly the composite map \(\tilde{{\mathbf {F}}}(\hat{{\mathbf {x}}}) = {\mathbf {F}}_{\kappa ^-}^{-1}({\mathbf {F}}_{\kappa ^+}(\hat{{\mathbf {x}}}))\). That is
$$\begin{aligned} \tilde{{\mathbf {F}}}(\hat{{\mathbf {x}}}) = {\mathbf {J}}^{-1}_{\kappa ^-}( {\mathbf {J}}_{\kappa ^+} \hat{{\mathbf {x}}} + {\mathbf {t}}_{\kappa ^+} ) - {\mathbf {J}}^{-1}_{\kappa ^-} {\mathbf {t}}_{\kappa ^-} = \underbrace{{\mathbf {J}}^{-1}_{\kappa ^-}{\mathbf {J}}_{\kappa ^+}}_{\tilde{{\mathbf {J}}}} \hat{{\mathbf {x}}} + \underbrace{{\mathbf {J}}^{-1}_{\kappa ^-} ({\mathbf {t}}_{\kappa ^+} - {\mathbf {t}}_{\kappa ^-})}_{\tilde{{\mathbf {t}}}}, \end{aligned}$$
where the matrix \(\tilde{{\mathbf {J}}}\) is diagonal since \({\mathbf {J}}^{-1}_{\kappa ^-}\) and \({\mathbf {J}}_{\kappa ^+}\) are diagonal. We then haveCombining (30) and (31), and denoting by \(\hat{F}^+ = {\mathbf {F}}_{\kappa ^+}^{-1}(F)\), cf. Fig. 11, from (29) we obtainwhere Open image in new window and Open image in new window are defined asHere, as before, \(C_{i,n}\) are the coefficients of the homogeneous function expansion of the Legendre polynomials in \((-1,1)\), while \(\tilde{X}_{j,m}\) and \(\tilde{Y}_{j,m}\) are defined by
$$\begin{aligned} \left. \begin{aligned} \tilde{X}_{j,m}&= \sum _{r = m}^j C_{j,r}\ {\left( {\begin{array}{c}r\\ m\end{array}}\right) }\ (\tilde{{\mathbf {J}}}_{1,1})^m\ (\tilde{{\mathbf {t}}}_1)^{r-m} \\ \tilde{Y}_{j,m}&= \sum _{r = m}^j C_{j,r}\ {\left( {\begin{array}{c}r\\ m\end{array}}\right) }\ (\tilde{{\mathbf {J}}}_{2,2})^m\ (\tilde{{\mathbf {t}}}_2)^{r-m} \end{aligned}\right\} \quad \text {for } 0 \le m \le p_{\kappa ^-},\ m \le j \le p_{\kappa ^-}; \end{aligned}$$
here, we have exploited the Newton-binomial expansion of the terms \((\tilde{{\mathbf {J}}}_{1,1} \hat{x} + \tilde{{\mathbf {t}}}_1)^k\) and \((\tilde{{\mathbf {J}}}_{2,2} \hat{y} + \tilde{{\mathbf {t}}}_2)^l\) appearing in equation (31).
Fig. 11

Example of a polygonal elements Open image in new window , together with the bounded boxes \(B_{\kappa ^{\pm }}\), and the local maps \({\mathbf {F}}_{\kappa ^{\pm }}: \hat{\kappa } \rightarrow \kappa ^{\pm }\) for the common face \(F \subset \kappa ^{\pm }\)

Similar considerations allow us to computewhere \({\mathcal {C}}''_{i,j,k}\) are defined as
$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {C}}''_{0,j,k} = 0 &{} \forall j,\ \forall k,\\ {\mathcal {C}}''_{i,j,k} = \sum _{n + m = k} C'_{i,n}C_{j,m},\ &{} 1 \le i \le p_{\kappa ^+},0 \le j \le p_{\kappa ^+},0 \le k \le i+j-1, \end{array}\right. } \end{aligned}$$
and where \({\mathbf {n}}^+ = [n_x^+, n_y^+]^\top \) is the unit outward normal vector to the physical face F from \(\kappa ^+\). Similarly,where we have also introduced Open image in new window and Open image in new window defined as

Remark 5

The coefficients \(\tilde{X}\) and \(\tilde{Y}\) depend on the maps \({\mathbf {F}}_{\kappa ^+}\) and \({\mathbf {F}}_{\kappa ^-}\), as well as Open image in new window and Open image in new window ; thereby, they must be computed for each element \(\kappa \) in the mesh Open image in new window .

Remark 6

With regards the computation of the forcing term
$$\begin{aligned} {\mathbf {f}}_i = \int _{\varOmega } f({\mathbf {x}}) \phi _i({\mathbf {x}}) \mathrm {d}{\mathbf {x}},\ \forall i=1,\ldots ,N_h, \end{aligned}$$
(32)
we point out that the quadrature method proposed in this paper allows to exactly evaluate (32) when f is a constant or a polynomial function. If f is a general function, an explicit polynomial approximation of f is required.

5 Numerical Experiments

We present some two- and three-dimensional numerical experiments to test the practical performance of the proposed approach. Here, the results are compared with standard assembly algorithms based on employing efficient quadrature rules on a sub-tessellation.

5.1 Two-Dimensional Test Case

We test the performance of the algorithm outlined in Sect. 4 for the computation of the elemental mass and stiffness matrices resulting from the DG discretization (19) on Voronoi decompositions as shown in Fig. 12. In particular, we compare the CPU-time needed to assemble the local and global elemental matrices using Algorithm 2, cf. Sect. 4, with Quadrature Integration over polygonal domains, based on the sub-tessellation method on polygons and Gaussian line integration for the related interface terms. More precisely, given Open image in new window , the sub-tessellation scheme on \(\kappa \) is performed by constructing a non-overlapping sub-tessellation \(\kappa _{{\mathscr {S}}} = \{ \tau _{\kappa } \}\) consisting of standard triangular elements; in particular, as, for our tests, we consider Voronoi numerical grids, we exploit the convexity of \(\kappa \) and define \(\kappa _{{\mathscr {S}}}\) by connecting the centre of mass of \(\kappa \) with its vertices. As an example, if we consider computing the elemental mass matrix \({\mathbf {M}}_{i,j}^{\kappa }\), we have that
$$\begin{aligned} {\mathbf {M}}_{i,j}^{\kappa } = \int _{\kappa } \phi _i \phi _j \mathrm {d} {\mathbf {x}} \approx \sum _{\tau _{\kappa } \in \kappa _{\mathscr {S}}} \sum _{r = 1}^{q_{\tau _{\kappa }}} \phi _i({\mathbf {F}}_{\tau _\kappa }(\varvec{\xi }_r)) \phi _j({\mathbf {F}}_{\tau _\kappa }(\varvec{\xi }_r)) | {\mathbf {J}}_{\tau _\kappa }| \omega _r, \end{aligned}$$
where \({\mathbf {F}}_{\tau _\kappa }: \hat{\tau } \rightarrow \tau _\kappa \) is the mapping from the reference simplex \(\hat{\tau }\) to \(\tau _\kappa \), with Jacobian \(|{\mathbf {J}}_{\tau _\kappa }|\), and \(\{ (\varvec{\xi }_r, \omega _r) \}_{r=1}^{q_{\kappa }}\) denotes the quadrature rule defined on \(\hat{\tau }\). The construction of quadrature rules on \(\hat{\tau }\) may be computed based on employing the Duffy transformation, whereby the reference tensor-product element \((-1,1)^2\) is mapped to the reference simplex. As the algorithm outlined in Sect. 4 does not require the definition of quadrature nodes and weights, in the following we will refer to it as the Quadrature Free Method. Consider the problem (19) introduced in Sect. 3 with \(d=2\) and \(\varOmega = (0,1)^2\), where we select the set of basis functions \(\{ \phi _i \}_{i=1}^{N_h}\) for \(V_h\) as described in Sect. 4. In order to quantify the performance of the proposed approach, we consider a series of numerical tests obtained by varying the polynomial degree \(p_{\kappa }=p\) for all Open image in new window , between 1 and 6 and by employing a series of uniform polygonal meshes of different granularity, cf. Fig. 12. The numerical grids are constructed based on employing PolyMesher, cf. [65]. Here, we are interested in the CPU time needed to assemble the matrices (21) and (22).
Fig. 12

Example of Voronoi mesh on \(\varOmega =(0,1)^2\). a 50 elements. b 250 elements. c 1000 elements

Fig. 13

Comparison of the CPU time needed to assemble the global matrices \({\mathbf {M}}\) and \({\mathbf {V}}\) for a two-dimensional problem by using the proposed quadrature free method and the classical sub-tessellation scheme. For each algorithm, each line is obtained by fixing the polynomial approximation degree \(p \in \{1,2,3\}\) (left) and \(p \in \{4,5,6\}\) (right), and measuring the CPU time by varying the number of elements in the underlying mesh. a Comparison for \(p \in \{1,2,3\}\). b Comparison for \(p \in \{4,5,6\}\)

In the first test case, we consider the CPU time needed to assemble the matrices \({\mathbf {M}}\) and \({\mathbf {V}}\). As pointed out in Sect. 4, these matrices are block diagonal and each block consists of an integral over each polygonal element Open image in new window . In Fig. 13 we present the comparison between the CPU times needed to assemble the global matrices \({\mathbf {M}}\) and \({\mathbf {V}}\) based on employing the quadrature free method and quadrature integration (based on sub-tessellation) when varying the number of elements \(N_{e} \in \{ 64, 256, 1024, 4094, 16384, 65536\}\) and the polynomial degree \(p \in \{ 1,2,3\}\) (left), and \(p \in \{4,5,6\}\) (right). Clearly, our approach outperforms the classical sub-tessellation method leading to substantial gains in efficiency. For a more detailed comparison, we have presented in Fig. 15a the logarithmic-scaled graphs of each computation: from the results of Fig. 15a we observe that the CPU time grows like \(\mathcal{O}(N_e)\), as \(N_e\) increases, as expected.
Fig. 14

Comparison of the CPU time needed to assemble the global matrices \({\mathbf {S}}\) and \({\mathbf {I}}\) for a two-dimensional problem by using the proposed quadrature free method and the classical Gauss line integration scheme. For each approach, each line is obtained by fixing the polynomial approximation degree \(p \in \{1,2,3\}\) (left) and \(p \in \{4,5,6\}\) (right), and measuring the CPU time by varying the number of elements in the underlying mesh. a Comparison for \(p \in \{1,2,3\}\). b Comparison for \(p \in \{4,5,6\}\)

Fig. 15

Comparison between the CPU time needed by the two method to assemble the global matrices \({\mathbf {M}}\) and \({\mathbf {V}}\) (left) and \({\mathbf {S}}\) and \({\mathbf {I}}\) (right) for a three-dimensional problem, versus the number of elements and for different choices of \(p=3,\ldots ,6\) (log–log scale). a CPU time comparison needed to assemble \({\mathbf {M}}\) and \({\mathbf {V}}\) in log–log scale. b CPU time comparison needed to assemble \({\mathbf {S}}\) and \({\mathbf {I}}\) in log-log scale

We have repeated the same set of numerical experiments measuring the CPU times needed to assemble the face terms appearing in the matrices \({\mathbf {S}}\) and \({\mathbf {I}}\); these results are reported in Fig. 14. Here, the domains of integration of the integrals involved are the edges of the polygonal elements, which are simply line segments in the plane \({\mathbf {R}}^2\). Here, we compare the quadrature free method described in Sect. 4.3 with classical Gaussian line integration, where the integrating function is pointwise evaluated on the physical numerical nodes lying on each face. The graphs in Fig. 14a, b show the comparison between the CPU time taken for the two different approaches. Here, we again observe that significant computational savings are made when the proposed quadrature free method is employed, though the increase in efficiency is less than that attained for the computation of the volume integrals. In Fig. 15b we plot the logarithmic-scaled CPU time with respect to the number of mesh elements; again the CPU time grows as \(\mathcal{O}(N_e)\) as \(N_e\) increases.

Referring to Figs. 13 and 14, we observe that the cost of assembly of the matrices \({\mathbf {M}}\) and \({\mathbf {V}}\), which involve volume integrals over each element \(\kappa \) in the computational mesh Open image in new window , is more expensive than the time it takes to assemble the face-based matrices \({\mathbf {S}}\) and \({\mathbf {I}}\), when the classical Gaussian line integration method is employed. This is, of course, due to the greater number of function evaluations required to compute \({\mathbf {M}}\) and \({\mathbf {V}}\) on the underlying sub-tessellation; note that in two-dimensions, a sub-tesellation of the faces is not necessary, since they simply consist of line segments. However, the opposite behaviour is observed when the quadrature free method is employed; in this case, the volume integrals can be very efficiently computed since the coefficients \(\mathcal{C}_{i,j,k}\) and \(\mathcal{C}'_{i,j,k}\) only need to be computed once, cf. Sect. 4.2. On the other hand, computing the face integrals present in \({\mathbf {S}}\) and \({\mathbf {I}}\) requires the evaluation of the coefficients Open image in new window , and Open image in new window , cf. Sect. 4.3, which must be computed for each face Open image in new window .

5.2 Three-Dimensional Test Case

We now consider the diffusion–reaction problem (19) with \(d=3\) and \(\varOmega = (0,1)^3\). The polyhedral grids employed for this test case are defined by agglomeration: starting from a fine partition Open image in new window of \(\varOmega \) consisting of \(N_{fine}\) disjoint tetrahedrons \(\{ \kappa _{f}^i\}_{i=1}^{N_{fine}}\), such that \(\overline{\varOmega } = \cup _{i=1}^{N_{fine}} \overline{\kappa }_{f}^i\), a coarse mesh Open image in new window of \(\varOmega \) consisting of disjoint polyhedral elements \(\kappa \) can be defined such thatwhere Open image in new window denotes the set of fine elements which forms \(\kappa \). Here, the agglomeration of fine tetrahedral elements is performed based on employing the METIS library for graph partitioning, cf., for example, [45, 46]. With this definition each polyhedral element is typically non-convex. For simplicity, we have considered only the case of simply connected elements. In this particular case, the faces of the mesh Open image in new window are the triangular intersections of two-dimensional facets of neighbouring elements. Figure 16 shows three examples of the polyhedral elements resulting from agglomeration.
Fig. 16

Example of polyhedral elements Open image in new window obtained by agglomeration of tetrahedrons. \(\kappa _1\) has 18 triangular faces, \(\kappa _2\) has 20 triangular faces and \(\kappa _3\) has 22 triangular faces. a Element \(\kappa _1\). b Element \(\kappa _2\). c Element \(\kappa _3\)

Fig. 17

Comparison of the CPU time needed to assemble the global matrices \({\mathbf {M}}\) and \({\mathbf {V}}\) for a three-dimensional problem by using the proposed quadrature free method and the classical sub-tessellation method. For each approach, each line is obtained by fixing the polynomial approximation degree \(p \in \{1,2,3\}\) (left) and \(p \in \{4,5,6\}\) (right), and measuring the CPU time by varying the number of elements of the underlying mesh. a Comparison for \(p \in \{1,2,3\}\). b Comparison for \(p \in \{4,5,6\}\)

Fig. 18

Comparison of the CPU time needed to assemble the global matrices \({\mathbf {S}}\) and \({\mathbf {I}}\) for a three–dimensional problem by using the proposed quadrature free method and the classical sub-tessellation method. For each approach, each line is obtained by fixing the polynomial approximation degree \(p \in \{1,2,3\}\) (left) and \(p \in \{4,5,6\}\) (right), and measuring the CPU time by varying the number of elements of the underlying mesh. a Comparison for \(p \in \{1,2,3\}\). b Comparison for \(p \in \{4,5,6\}\)

Fig. 19

Comparison between the CPU time needed by the two method to assemble the global matrices \({\mathbf {M}}\) and \({\mathbf {V}}\) (left) and \({\mathbf {S}}\) and \({\mathbf {I}}\) (right) for a three-dimensional problem, versus the number of elements and for different choices of \(p=3,\ldots ,6\) (log–log scale). a CPU time comparison needed to assemble \({\mathbf {M}}\) and \({\mathbf {V}}\) in log–log scale. b CPU time comparison needed to assemble \({\mathbf {S}}\) and \({\mathbf {I}}\) in log–log scale

We perform a similar set of experiments as the ones outlined in Sect. 5.2 for the two-dimensional case. Again, we compare the CPU time required by the proposed quadrature free method with the quadrature integration/sub-tessellation approach to assemble the stiffness and mass matrices resulting from the DG discretization of problem (19). Numerical integration over a polyhedral domain is required to assemble the matrices \({\mathbf {M}}\) and \({\mathbf {V}}\), cf. (21), whereas for the computation of \({\mathbf {S}}\) and \({\mathbf {I}}\), cf. (22), a cubature rule over polygonal faces (here triangular shaped) is needed. In general, for three-dimensional problems the quadrature integration approach consists in the application of the sub-tessellation method both for volume and face integrals, although in this particular case no sub-tessellation is required for the face integrals, since they simply consist of triangular domains. Moreover, as in this case a sub-tessellation into tetrahedral domains is already given by the definition of the polyhedral mesh, the quadrature integration for volume integrals on a general agglomerated polyhedral element \(\kappa = \cup _{\kappa _{f}' \in {\mathscr {S}}_{\kappa }} \kappa _{f}'\) is realized by applying an exact quadrature rule on each tetrahedron \(\kappa _{f}' \in {\mathscr {S}}_{\kappa }\). The comparison of the CPU times for the two methods outlined here are presented for a set of agglomerated polyhedral grids where we vary the number of elements \(N_e \in \{ 5, 40, 320, 2560, 20480\}\), and the polynomial degree \(p \in \{ 1,2,3,4,5,6\}\). For each agglomerated polyhedral grid Open image in new window we have chosen the corresponding fine tetrahedral grid Open image in new window such that the cardinality of the set \({\mathscr {S}}_{\kappa }\) appearing in (33) is Open image in new window . The results are shown in Fig. 17 for the computation of the matrices \({\mathbf {M}}\) and \({\mathbf {V}}\), and in Fig. 18 for the computation of matrices \({\mathbf {S}}\) and \({\mathbf {I}}\). Here, we observe analogous behaviour to the two-dimensional case: the quadrature free method substantially outperforms quadrature integration both for the computation of the volume and face integrals. We also have reported in Fig. 19 the logarithmic-scaled graphs of each computation, showing that, as expected, the gain in terms of CPU time attained by exploiting the proposed method is more evident here, with respect to the two-dimensional case, also for the face integrals.

6 Conclusions

In this article we have proposed a new approach for the numerical evaluation of the integrals required to assemble the mass and stiffness matrices arising from the DG discretization of diffusion–reaction problems, where the underlying mesh is composed of polygonal/polyhedral elements. Starting from the idea proposed in [27] for the integration of homogeneous functions, we have developed a cubature method which does not require the definition of a set of nodes and weights on the domain of integration, and allows for the exact integration of polynomial functions based on evaluating the integrand and its derivatives only at the vertices of the polytopal integration domain. This approach shows a remarkable gain in terms of CPU time with respect to classical quadrature rules, maintaining the same degree of accuracy. On the one hand, the number of computations is optimized, with respect to the polynomial degree of the integrand, and moreover less memory storage is required as no sub-tessellation and quadrature nodes and weights are required. With regards the three-dimensional tests presented in Sect. 5.2, we note that more substantial gains in terms of CPU time, with respect to classical approaches, can be obtained if the underlying grid is composed of pure (not agglomerated) polyhedral elements: firstly, this is because a sub-partition should be defined on the fly for each element, and secondly, as faces are not only triangles but possibly polygons of arbitrary shape, a sub-tessellation is needed also for surface integrals. The proposed technique is completely general and can be extended to several numerical methods based on discrete spaces defined on polygonal/polyhedral meshes, such as Virtual Element Methods, Mimetic Finite Differences, Hybrid High-Order Methods, Hybridizable Discontinuous Galerkin Schemes, and Polygonal Finite Element Methods, for example. We stress that for moderate polynomial degrees, the proposed integration technique, which involves exact integration of bivariate and trivariate functions in two- and three-dimensions, respectively, has been observed to be numerically stable.

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Authors and Affiliations

  1. 1.MOX-Laboratory for Modeling and Scientific Computing, Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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