Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Abstract

Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order $k + 1$ to order $2k + 1$. Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577–606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-Sánchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results.

Appendix

In this appendix, we provide some details for the box-spline-based formulation of hex splines. This formulation provides a compact and efficient mechanism to evaluate and study the properties of higher order hex splines. We start the discussion with a brief introduction to box splines as one of the most generic spline functions.

A box spline $\mathbb {R}^d$ is defined by a set of n vectors $\xi _1, \ldots , \xi _n \in \mathbb {R}^d$. Geometrically, a box spline is the shadow of a hypercube in $\mathbb {R}^n$ that has been projected onto $\mathbb {R}^d$ where $n > d$. Each $\xi$ denotes the shadow of an edge of the hypercube in $\mathbb {R}^n$. The simplest box spline in $\mathbb {R}^d$ corresponds to the case where $d=n$. In this case, a box spline is defined as the (normalized) indicator function of the parallelepiped formed by the d vectors in $\mathbb {R}^d$.

$$\begin{aligned} M_{\varXi }(\mathbf {x}) = \left\{ \begin{array}{l l} \frac{1}{|\det (\varXi )|} &{} \quad \mathbf {x} = {\sum }_{i=1}^d t_i \xi _i \text { for } 0 \le t_i \le 1 \\ 0 &{} \quad \text {otherwise} \end{array} \right. , \end{aligned}$$

(30)

where $\varXi :=[\xi _1, \ldots , \xi _n]$ denotes the matrix of directions. For $n > d$, a box spline can be defined recursively as

$$\begin{aligned} M_{\varXi \cup \xi _k} (\mathbf {x}) = \int _0^1 M_{\varXi }(\mathbf {x}-t \mathbf {\xi }) dt, \end{aligned}$$

(31)

where $\varXi \cup \xi _k$ denotes the addition of $\xi _k$ to the matrix of direction $\varXi \cup \xi _k := [\xi _1,\cdots , \xi _n, \xi _k]$. The convolution in the relation above can be considered as smearing the original box spline $M_{\varXi }$ along the new direction $\xi _k$. Consequently, the support of the new box spline $M_{\varXi \cup \xi _k}$ can be considered as the Minkowski sum of the vectors in $\varXi \cup \xi _k$. Figure 13 demonstrates the idea behind the convolution in Eq. 31 graphically. Similarly, the convolution of two box splines together results in a new box spline

$$\begin{aligned} M_{\varXi _1} *M_{\varXi _2} = M_{[\varXi _1 \cup \varXi _2]}. \end{aligned}$$

(32)

Fig. 13

The support of $M_{[\xi _1, \xi _2]}$ (a). The support of $M_{[\xi _1, \xi _2, \xi _3]}$ can be considered as smearing the support of $M_{[\xi _1, \xi _2]}$ along $\xi _3$ (b). The support of $M_{[\xi _1, \xi _2, \xi _3]}$ can be considered as the Minkowski sum of $\xi _1, \xi _2$, and $\xi _3$ (c)

Note that the vectors can appear with some multiplicity in the matrix of directions of a box spline.

Fig. 14

First-order hex spline can be decomposed into three box splines

The matrix of directions fully specifies all the properties of $M_{\varXi }$ but the order of the vectors in $\varXi$ does not have any effect on its properties. For instance, let $\kappa$ denote the minimum number of directions whose removal from $\varXi$ makes the remaining directions not span $\mathbb {R}^d$. The value of $\kappa$ specifies the order of continuity of a box spline. That is, a box spline formed by $\varXi$ is $C^{\kappa -2}$ continuous [9]. A box spline is a piecewise polynomial function and can be efficiently evaluated using the recursive definition in Eq. 30 or using a Fourier transform analysis [9]. It is worth mentioning that the derivatives of $M_{\varXi }$ in any direction $Z \in \varXi$ can be exactly evaluated using a differencing operator

$$\begin{aligned} D_{Z} M_{\varXi } = \partial _{Z} M_{\varXi \backslash Z} \text { for } Z \subseteq \varXi , \end{aligned}$$

(33)

where $\varXi \backslash Z$ denotes removal of direction Z from the matrix of direction and $\partial _{Z}$ denotes a backward difference operator in the direction of Z. In general, the derivative of $M_{\varXi }$ in any arbitrary direction can still be rewritten in terms of (a linear combination of) the derivatives in the direction Z where $Z \in \varXi$ [9, Lemma 34]. The interested reader can consult [9] for a thorough discussion of box splines and it properties.

Both B-splines and hex splines can be written in terms of box splines. For example, the first-order hex spline can be written and evaluated as a summation of the indicator function of three parallelepipeds that constitute the Voronoi cell of the hexagonal lattice (i.e., the hexagon presented in Fig. 1b). Each parallelepiped can be represented using a box spline

$$\begin{aligned} \eta _1 = \frac{1}{3}\big (M_{[\xi _1, \xi _2]} + M_{[\xi _1, \xi _3]} + M_{[\xi _2, \xi _3]} \big ), \end{aligned}$$

(34)

where $\xi _i$ represents the $i^{th}$ column of matrix $\mathbf {H}_3$. Figure 14 demonstrates the decomposition of the first-order hex spline into three box splines. We can now use the box spline convolution rule in order to write the higher order hex splines as

$$\begin{aligned} \eta _{n} = \big ( \eta _1 \big )^{*n} = \frac{1}{3^n} \big ( M_{[\xi _1, \xi _2]} + M_{[\xi _1, \xi _3]} + M_{[\xi _2, \xi _3]} \big )^{*n}, \end{aligned}$$

(35)

where $( \eta _1)^{*n}$ is a short-hand notation for n-times self-convolution.

For example, the second-order hex spline can be written in terms of six box splines

$$\begin{aligned} \begin{aligned} \eta _2&= \frac{1}{3^2} \big ( M_{[\xi _1, \xi _2]} + M_{[\xi _1, \xi _3]} + M_{[\xi _2, \xi _3]} \big )^{*2} \\&= \frac{1}{9} \big ( M_{[\xi _1, \xi _1, \xi _2, \xi _2]} + M_{[\xi _1, \xi _1, \xi _3, \xi _3]} + M_{[\xi _2, \xi _2, \xi _3, \xi _3]} \big ) \\&\quad + \frac{2}{9} \big ( M_{[\xi _1, \xi _1, \xi _2, \xi _3]} + M_{[\xi _1, \xi _2, \xi _2, \xi _3]} + M_{[\xi _1, \xi _2, \xi _3, \xi _3]} \big ). \end{aligned} \end{aligned}$$

(36)

Not that unlike B-splines, the derivatives of a hex spline cannot be written in terms of lower order hex splines. However, the derivatives of hex splines can be written compactly in terms of the derivative of its constituent box splines (see Eq. 33), and be exactly evaluated using difference operators [9, Lemma 34].