# Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

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## 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.

## Keywords

B-splines Hex splines Box splines Smoothness-Increasing Accuracy-Conserving (SIAC) filtering Quasi-interpolation Approximation theory Discontinuous Galerkin## Notes

### Acknowledgements

This work was sponsored in part by the Air Force Office of Scientific Research (AFOSR), Computational Mathematics Program (Program Manager: Dr. Jean-Luc Cambier), under Grant Nos. FA9550-12-1-0428 (first and fourth author), FA8655-13-1-3017 (third author). The second and fourth authors are sponsored in part by the Army Research Office (Program manager: Dr. Mike Coyle) under Grant No. W911NF-15-1-0222. In addition, the authors would like to acknowledge the anonymous reviewers for providing comments that improved the manuscript.

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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