deal.II Implementation of a Weak Galerkin Finite Element Solver for Darcy Flow

Wang, Zhuoran; Harper, Graham; O’Leary, Patrick; Liu, Jiangguo; Tavener, Simon

doi:10.1007/978-3-030-22747-0_37

deal.II Implementation of a Weak Galerkin Finite Element Solver for Darcy Flow

Conference · Sat Jun 08 00:00:00 EDT 2019

DOI:https://doi.org/10.1007/978-3-030-22747-0_37· OSTI ID:1594093

Wang, Zhuoran ^[1]; Harper, Graham ^[2]; O’Leary, Patrick ^[3]; Liu, Jiangguo ^[2]; Tavener, Simon ^[2]

Colorado State Univ., Fort Collins, CO (United States); Kitware, Inc.
Colorado State Univ., Fort Collins, CO (United States)
Kitware, Inc., Clifton Park, NY (United States)

This paper presents a weak Galerkin (WG) finite element solver for Darcy flow and its implementation on the deal.II platform. The solver works for quadrilateral and hexahedral meshes in a unified way. It approximates pressure by Q-type degree $k(≥ 0)$ polynomials separately defined in element interiors and on edges/faces. In this work, numerical velocity is obtained in the unmapped Raviart-Thomas space $$RT_{[k]}$$ via postprocessing based on the novel concepts of discrete weak gradients. The solver is locally mass-conservative and produces continuous normal fluxes. The implementation in deal.II allows polynomial degrees up to 5. Numerical experiments show that our new WG solver performs better than the classical mixed finite element methods.

View Conference

Research Organization:: Kitware, Inc., Clifton Park, NY (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Chemical Sciences, Geosciences & Biosciences Division; National Science Foundation (NSF)

DOE Contract Number:: SC0019609

OSTI ID:: 1594093

Resource Type:: Conference paper/presentation

Conference Information:: International Conference on Computational Science, Faro (Portugal), 12-14 Jun 2019; From, proceedings, part IV

Country of Publication:: United States

Language:: English

References (18)

Superconvergence and H(div) projection for discontinuous Galerkin methods: SUPERCONVERGENCE AND H(DIV) PROJECTION FOR DG Bastian, Peter; Rivière, Béatrice International Journal for Numerical Methods in Fluids, Vol. 42, Issue 10 https://doi.org/10.1002/fld.562	journal	July 2003
The Mathematical Theory of Finite Element Methods Brenner, Susanne C.; Scott, L. Ridgway Texts in Applied Mathematics https://doi.org/10.1007/978-0-387-75934-0	book	January 2008
Mixed and Hybrid Finite Element Methods Brezzi, Franco; Fortin, Michel Springer Series in Computational Mathematics https://doi.org/10.1007/978-1-4612-3172-1	book	January 1991
A weak Galerkin finite element method for second-order elliptic problems Wang, Junping; Ye, Xiu Journal of Computational and Applied Mathematics, Vol. 241 https://doi.org/10.1016/j.cam.2012.10.003	journal	March 2013
A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods Lin, Guang; Liu, Jiangguo; Sadre-Marandi, Farrah Journal of Computational and Applied Mathematics, Vol. 273 https://doi.org/10.1016/j.cam.2014.06.024	journal	January 2015
A weak Galerkin finite element method with polynomial reduction Mu, Lin; Wang, Junping; Ye, Xiu Journal of Computational and Applied Mathematics, Vol. 285 https://doi.org/10.1016/j.cam.2015.02.001	journal	September 2015
Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity Lin, Guang; Liu, Jiangguo; Mu, Lin Journal of Computational Physics, Vol. 276 https://doi.org/10.1016/j.jcp.2014.07.001	journal	November 2014
The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes Liu, Jiangguo; Tavener, Simon; Wang, Zhuoran Journal of Computational Physics, Vol. 359 https://doi.org/10.1016/j.jcp.2018.01.001	journal	April 2018
DarcyLite: A Matlab Toolbox for Darcy Flow Computation Liu, Jiangguo; Sadre-Marandi, Farrah; Wang, Zhuoran Procedia Computer Science, Vol. 80 https://doi.org/10.1016/j.procs.2016.05.485	journal	January 2016
The THex Algorithm and a Simple Darcy Solver on Hexahedral Meshes Harper, Graham; Liu, Jiangguo; Zheng, Bin Procedia Computer Science, Vol. 108 https://doi.org/10.1016/j.procs.2017.05.148	journal	January 2017
An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST) Lie, Knut-Andreas https://doi.org/10.1017/9781108591416	book	December 2018
Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities Durlofsky, Louis J. Water Resources Research, Vol. 30, Issue 4 https://doi.org/10.1029/94WR00061	journal	April 1994
Locally Conservative Fluxes for the Continuous Galerkin Method Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Wang, Haiying SIAM Journal on Numerical Analysis, Vol. 45, Issue 4 https://doi.org/10.1137/060666305	journal	January 2007
A Locally Conservative Finite Element Method Based on Piecewise Constant Enrichment of the Continuous Galerkin Method Sun, Shuyu; Liu, Jiangguo SIAM Journal on Scientific Computing, Vol. 31, Issue 4 https://doi.org/10.1137/080722953	journal	January 2009
On the Application of the Continuous Galerkin Finite Element Method for Conservation Problems Bush, L.; Ginting, V. SIAM Journal on Scientific Computing, Vol. 35, Issue 6 https://doi.org/10.1137/120900393	journal	January 2013
Two Families of $H$(div) Mixed Finite Elements on Quadrilaterals of Minimal Dimension Arbogast, Todd; Correa, Maicon R. SIAM Journal on Numerical Analysis, Vol. 54, Issue 6 https://doi.org/10.1137/15M1013705	journal	January 2016
Lowest-Order Weak Galerkin Finite Element Method for Darcy Flow on Convex Polygonal Meshes Liu, Jiangguo; Tavener, Simon; Wang, Zhuoran SIAM Journal on Scientific Computing, Vol. 40, Issue 5 https://doi.org/10.1137/17M1145677	journal	January 2018
deal.II---A general-purpose object-oriented finite element library Bangerth, W.; Hartmann, R.; Kanschat, G. ACM Transactions on Mathematical Software, Vol. 33, Issue 4 https://doi.org/10.1145/1268776.1268779	journal	August 2007

Similar Records

Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity

Journal Article · Sat Nov 01 00:00:00 EDT 2014 · Journal of Computational Physics · OSTI ID:1236930

Related Subjects

54 ENVIRONMENTAL SCIENCES
and Weak Galerkin
Darcy flow
deal.II
Finite element methods
Hexahedral meshes
Quadrilateral meshes

deal.II Implementation of a Weak Galerkin Finite Element Solver for Darcy Flow

Citation Formats

References (18)

Similar Records

Related Subjects