Browse > Article
http://dx.doi.org/10.12941/jksiam.2021.25.066

REVIEW AND IMPLEMENTATION OF STAGGERED DG METHODS ON POLYGONAL MESHES  

KIM, DOHYUN (SCHOOL OF MATHEMATICS AND COMPUTING (COMPUTATIONAL SCIENCE AND ENGINEERING), YONSEI UNIVERSITY)
ZHAO, LINA (DEPARTMENT OF MATHEMATICS, CITY UNIVERSITY OF HONG KONG)
PARK, EUN-JAE (SCHOOL OF MATHEMATICS AND COMPUTING (COMPUTATIONAL SCIENCE AND ENGINEERING), YONSEI UNIVERSITY)
Publication Information
Journal of the Korean Society for Industrial and Applied Mathematics / v.25, no.3, 2021 , pp. 66-81 More about this Journal
Abstract
In this paper, we review the lowest order staggered discontinuous Galerkin methods on polygonal meshes in 2D. The proposed method offers many desirable features including easy implementation, geometrical flexibility, robustness with respect to mesh distortion and low degrees of freedom. Discrete function spaces for locally H1 and H(div) spaces are considered. We introduce special properties of a sub-mesh from a given star-shaped polygonal mesh which can be utilized in the construction of discrete spaces and implementation of the staggered discontinuous Galerkin method. For demonstration purposes, we consider the lowest case for the Poisson equation. We emphasize its efficient computational implementation using only geometrical properties of the underlying mesh.
Keywords
Staggered grid; Discontinuous Galerkin method; Lowest order methods; Polygonal Meshes; Implementation; Static condensation;
Citations & Related Records
연도 인용수 순위
  • Reference
1 E.L. Wachspress, A Rational Finite Element Basis, Academic Press, New York, New York, 1975.
2 L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Mathematical Models and Methods in Applied Sciences, 23 (2013), 199-214.   DOI
3 D. A. Di Pietro and J. Droniou, The hybrid high-order method for polytopal meshes, Springer International Publishing, 2020.
4 J. Wang, and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, Journal of Computational and Applied Mathematics, 241 (2013), 103-115.   DOI
5 E. T. Chung and B. Engquist Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions, SIAM Journal on Numerical Analysis, 47 (2009), 3820-3848.   DOI
6 E. T. Chung, H. H. Kim and O.B. Widlund, Two-level overlapping Schwarz algorithms for a staggered discontinuous Galerkin method, SIAM Journal on Numerical Analysis, 51 (2013), 47-67.   DOI
7 H. H. Kim, E. T. Chung and C. S. Lee, A staggered discontinuous Galerkin method for the Stokes system, SIAM Journal on Numerical Analysis, 51 (2013), 3327-3350.   DOI
8 E. T. Chung, E.-J. Park and L. Zhao, Guaranteed a posteriori error estimates for a staggered discontinuous Galerkin method, Journal of Scientific Computing, 75 (2018), 1079-1101.   DOI
9 J.J. Lee and H. H. Kim, Analysis of a staggered discontinuous Galerkin method for linear elasticity, Journal of Scientific Computing, 66 (2016), 625-649.   DOI
10 L. Zhao and E.-J. Park, Fully computable bounds for a staggered discontinuous Galerkin method for the Stokes equations, Computers & Mathematics with Applications, 75 (2018), 4115-4134.   DOI
11 L. Zhao, E. T. Chung and M. Lam, A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits, Computer Methods in Applied Mechanics and Engineering, 364 (2020), Paper No.112986, 13 pp.
12 L. Zhao and E.-J. Park, A staggered discontinuous Galerkin method of minimal dimension on quadrilateral and polygonal meshes, SIAM Journal on Scientific Computing, 40 (2018), A2543-A2567.   DOI
13 L. Zhao, E.-J. Park and D.-w. Shin, A staggered DG method of minimal dimension for the Stokes equations on general meshes, Computer Methods in Applied Mechanics and Engineering, 345 (2019), 854-875.   DOI
14 L. Zhao and E.-J. Park, A lowest-order staggered DG method for the coupled Stokes-Darcy problem, IMA Journal of Numerical Analysis, 40 (2020), 2871-2897.   DOI
15 Y. Jeon, E.-J. Park, A hybrid discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 48 (5) (2010) 1968-1983.   DOI
16 Y. Jeon, E.-J. Park, New locally conservative finite element methods on a rectangular mesh, Numerische Mathematik, 123 (2013), no.1, 97-119.   DOI
17 Y. Jeon, and E.-J. Park, D.-w. Shin, Hybrid Spectral Difference Methods for an Elliptic Equation, Comput. Methods Appl. Math. vol.17 no.2 (2017), 253-267.   DOI
18 S. Yadav, A. Pani, and E.-J. Park, Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations, Math. Comp. 82 (2013), no. 283, 1297-1335.   DOI
19 P. F. Antonietti, S. Giani and P. Houston, hp-version composite discontinuous Galerkin methods for elliptic problems on complicated domains, SIAM Journal on Scientific Computing, 35 (2013), A1417A1439.   DOI
20 F. Bassi, L. Botti, A. Colombo, D. A. Di Pietro and P. Tesini, On the flexibility of agglomeration-based physical space discontinuous Galerkin discretizations, Journal of Compututational Physics, 231 (2012), 45-65.   DOI
21 L. Zhao and E.-J. Park, A priori and a posteriori error analysis for a staggered discontinuous Galerkin method for convection dominant diffusion equations, Journal of Computational and Applied Mathematics, 346 (2019), 63-83.   DOI
22 L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods on polytopal meshes, International Journal of Numerical Analysis and Modeling, 12 (2015), 31-53.
23 E. T. Chung and B. Engquist Optimal discontinuous Galerkin methods for wave propagation, SIAM Journal on Numerical Analysis, 44 (2006), 2131-2158.   DOI
24 E. T. Chung and W. Qiu, Analysis of an SDG method for the incompressible Navier-Stokes equations, SIAM Journal on Numerical Analysis, 55 (2017), 543-569.   DOI
25 L. Zhao, E. T. Chung, E.-J. Park, and G. Zhou, Staggered DG Method for Coupling of the Stokes and Darcy-Forchheimer Problems, SIAM Journal on Numerical Analysis 59 (2021), 1-31.   DOI
26 H. H. Kim, E. T. Chung and C.Y. Lam, Mortar formulation for a class of staggered discontinuous Galerkin methods, Computers & Mathematics with Applications, 71 (2016), 1568-1585.   DOI
27 E. T. Chung, P. Ciarlet Jr. and T. Yu, Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwells equations on Cartesian grids, Journal on Computational Physics, 235 (2013), 14-31.   DOI
28 E. T. Chung, C. Cockburn and G. Fu, The staggered DG method is the limit of a hybridizable DG method, SIAM Journal on Numerical Analysis, 52 (2014), 915-932.   DOI
29 L. Zhao and E.-J. Park, A new hybrid staggered discontinuous Galerkin method on general meshes, Journal of Scientific Computing, 82 (2020), Paper No.12, 33 pp.
30 L. Zhao and E.-J. Park, A staggered cell-centered DG method for linear elasticity on polygonal meshes, SIAM Journal on Scientific Computing, 42 (2020), A2158-A2181.   DOI
31 L. Zhao, E.-J. Park and E. T. Chung, Staggered discontinuous Galerkin methods for the Helmholtz equation with large wave number, Computers & Mathematics with Applications, 80 (2020), 2676-2690.   DOI
32 D. Kim, L. Zhao and E.-J. Park, Staggered DG methods for the pseudostress-velocity formulation of the Stokes equations on general meshes, SIAM Journal on Scientific Computing, 42 (2020), A2537-A2560.   DOI
33 D. A. Di Pietro, A. Ern and S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Computational Methods in Applied Mathematics, 14 (2014), 461-472.   DOI
34 T. Arbogast, C. N. Dawson, P. T. Keenan, M. F. Wheeler, and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM Journal on Scientific Computing, 19 (1998), 404-425.   DOI
35 L. Zhao, D. Kim, E.-J. Park and E. Chung, Staggered DG method with small edges for Darcy flows in fractured porous media, https://arxiv.org/abs/2005.10955.
36 Y. Jeon, E.-J. Park, D. Sheen, A hybridized finite element method for the Stokes problem, Computers & Mathematics with Applications Vol. 68, No. 12 Part B, (2014), 2222-2232.   DOI
37 M.-Y. KIM AND M. F. WHEELER, A multiscale discontinuous galerkin method for convection-diffusion-reaction problems, Computers & Mathematics with Applications, 68 (2014), pp. 2251-2261.   DOI
38 L. Beirao da Veiga, F. Brezzi, L. D. Marini and A. Russo The Hitchhiker's Guide to the Virtual Element Method, Mathematical Models and Methods in Applied Sciences, 24 (2014), 1541-1573.   DOI