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Locally Conservative Fluxes for the Continuous Galerkin Method

Bernardo Cockburn, Jayadeep Gopalakrishnan, Haiying Wang
2007 SIAM Journal on Numerical Analysis  
The standard continuous Galerkin (CG) finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations, a much needed quantity in many applications  ...  This leads to a conservative flux approximation with continuous normal components. This postprocessing applies for the CG method in its standard form or for a hybridized version of it.  ...  The authors would like to thank the reviewers for bringing to their attention the papers [7] and [26] . They would also like to thank Clint Dawson and Graham F.  ... 
doi:10.1137/060666305 fatcat:6kdgxhx3r5hc3fs327lghp5lkq

Page 88 of American Society of Civil Engineers. Collected Journals Vol. 128, Issue 1 [page]

2002 American Society of Civil Engineers. Collected Journals  
To “check” for local mass conservation, the individual face fluxes are summed and compared to the internal sources.  ...  Thus, intuitively, mass is “‘conserved” in some local sense for the discrete equations.  ... 

Partitions for Spectral (Finite) Volume Reconstruction in the Tetrahedron

Qian-Yong Chen
2005 Journal of Scientific Computing  
Numerical modeling of multi-phase flow in heterogeneous anisotropic porous media with locally conservative flux continuous schemes such as the multi-point flux methods, mixed finite volume methods, mixed  ...  Numerical solutions to conservation laws using high order methods such as spectral element methods, finite element discontinuous Galerkin methods, ENO/WENO Methods, and spectral volume methods.  ...  Society for Industrial and Applied Mathematics.  ... 
doi:10.1007/s10915-005-9009-0 fatcat:sg23yf6r5nb4dagwoymetaztq4

On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

Victor Ginting, Guang Lin, Jiangguo Liu
2015 Journal of Scientific Computing  
The numerical velocity obtained from solving the Darcy equation by the WGFEM is locally conservative and has continuous normal components across element interfaces.  ...  This paper presents studies on applying the novel weak Galerkin finite element method (WGFEM) to a two-phase model for subsurface flow, which couples the Darcy equation for pressure and a transport equation  ...  These two test examples confirm the merit of a locally conservative flux with continuous normal component that is obtained from using the weak Galerkin finite element method. modeling.  ... 
doi:10.1007/s10915-015-0021-8 fatcat:ap25wthkqran5h2drdkxbedaje

A Conservative DGM for Convection-Diffusion and Navier-Stokes Problems [chapter]

J. Tinsley Oden, Carlos Erik Baumann
2000 Lecture Notes in Computational Science and Engineering  
The variational formulation for diffusion terms produces a compact, locally conservative, higher-order accurate, and stable solver.  ...  An hp-adaptive con~ervative Discontinuous Galerkin Method for the solution of convection-diffusion problems is reviewed.  ...  Acknowledgement The support of this work by the Army Research Office under grant DAAH04-96-0062 is gratefully acknowledged.  ... 
doi:10.1007/978-3-642-59721-3_13 fatcat:42by7ntfxvgvpm2o5ihwrqelcq

Energy Stable and Momentum Conserving Hybrid Finite Element Method for the Incompressible Navier–Stokes Equations

Robert Jan Labeur, Garth N. Wells
2012 SIAM Journal on Scientific Computing  
Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh.  ...  Mass conservation, momentum conservation and global energy stability are proved for the time-continuous case, and for a fully discrete scheme.  ...  Local momentum conservation is in terms the numerical fluxσ h , as is a typical feature of discontinuous Galerkin methods.  ... 
doi:10.1137/100818583 fatcat:jzc6hec255eozadcowzq4nqzmi

A note on the least squares solution of the groundwater flow equation

L. R. Bentley
1994 Numerical Methods for Partial Differential Equations  
The least squares finite element method is a member of the weighted residuals class of numerical methods for solving partial differential equations.  ...  Solutions for problems containing parameters with large localized spatial gradients are characterized by errors that are propagated throughout the entire domain.  ...  The existence of large local gradients in parameter values leads to interelement interface problems, which are manifested as errors in the heads, flux, and local conservation of mass throughout the entire  ... 
doi:10.1002/num.1690100408 fatcat:nyitiwdexfcrritjo567mbwgta

DarcyLite: A Matlab Toolbox for Darcy Flow Computation

Jiangguo Liu, Farrah Sadre-Marandi, Zhuoran Wang
2016 Procedia Computer Science  
Specifically, four major types of finite element solvers, the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed finite element methods (MFEM), are examined  ...  This paper focuses on the finite element (FE) methods and the corresponding code modules in DarcyLite for solving the Darcy equation.  ...  element; (4) Local-mass-conservation residual on each element if a FE solver is not locally conservative; (5) Normal flux discrepancy across each interior edge if the normal fluxes are not continuous.  ... 
doi:10.1016/j.procs.2016.05.485 fatcat:my5fulzklnfwnfrggv67f34qwe

Page 2760 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
An advantage ascribed to the DGM is the local flux conservation property.  ...  First, error estimates for stabilized discontinuous Galerkin methods (SDGMs) are presented. Then, conservation laws are discussed for the DGM and CGM.  ... 

High-Order Finite Element Methods for Parallel Atmospheric Modeling [chapter]

Amik St.-Cyr, Stephen J. Thomas
2005 Lecture Notes in Computer Science  
High-order finite element methods for the atmospheric shallow water equations are reviewed.  ...  The accuracy and efficiency of nodal continuous and discontinuous Galerkin spectral elements are evaluated using the standard test problems proposed by Williamson et al (1992) .  ...  NCAR is supported by the National Science Foundation.  ... 
doi:10.1007/11428831_32 fatcat:alahj7kmfreilghz5qxaickbla

ON OPEN BOUNDARIES IN THE FINITE ELEMENT APPROXIMATION OF TWO-DIMENSIONAL ADVECTION-DIFFUSION FLOWS

FRANCISCO PADILLA, YVES SECRETAN, MICHEL LECLERC
1997 International Journal for Numerical Methods in Engineering  
Essentially, the space-time Crank-Nicolson-Galerkin formulation scheme was used to solve for a given conservative flow-field.  ...  With the proper choice of boundary conditions, the steady-state Galerkin and the transient Crank-Nicolson-Galerkin finite element schemes gave stable and precise results for advectiondominated transport  ...  ACKNOWLEDGEMENTS The authors are grateful to A. Khelifa, P. Boudreau and J.-L. Robert for their help and comments.  ... 
doi:10.1002/(sici)1097-0207(19970715)40:13<2493::aid-nme177>3.0.co;2-m fatcat:gwzuu5y5qzbmznjfo2mn6h5v3a

ON OPEN BOUNDARIES IN THE FINITE ELEMENT APPROXIMATION OF TWO‐DIMENSIONAL ADVECTION–DIFFUSION FLOWS

FRANCISCO PADILLA, YVES SECRETAN, MICHEL LECLERC
1997 International Journal for Numerical Methods in Engineering  
Essentially, the space-time Crank-Nicolson-Galerkin formulation scheme was used to solve for a given conservative flow-field.  ...  With the proper choice of boundary conditions, the steady-state Galerkin and the transient Crank-Nicolson-Galerkin finite element schemes gave stable and precise results for advectiondominated transport  ...  ACKNOWLEDGEMENTS The authors are grateful to A. Khelifa, P. Boudreau and J.-L. Robert for their help and comments.  ... 
doi:10.1002/(sici)1097-0207(19970715)40:13<2493::aid-nme177>3.3.co;2-d fatcat:nrz3xiqxfrfnxe5zjwys2szwh4

Conservation properties for the Galerkin and stabilised forms of the advection–diffusion and incompressible Navier–Stokes equations

Thomas J.R. Hughes, Garth N. Wells
2005 Computer Methods in Applied Mechanics and Engineering  
A common criticism of continuous Galerkin finite element methods is their perceived lack of conservation.  ...  As a result, conservative stabilised finite element procedures are presented for the advection-diffusion and incompressible Navier-Stokes equations.  ...  The authors would also like to express their appreciation to Victor Calo who read the manuscript thoroughly and made many helpful suggestions.  ... 
doi:10.1016/j.cma.2004.06.034 fatcat:rtxeoghvoffjnfrpkzrbcsb2rm

Local Time-Stepping for Explicit Discontinuous Galerkin Schemes [chapter]

Gregor Gassner, Michael Dumbser, Florian Hindenlang, Claus-Dieter Munz
2011 Computational Fluid Dynamics 2010  
Similarly for the discontinuous Galerkin method provided that the number of degrees of fredom is close to the number of nodes in computations with CD method.  ...  Abstract The present paper deals with the continuous work of extending the multidimensional limiting process (MLP), which has been quite successful in finite volume methods (FVM), into discontinuous Galerkin  ...  Acknowledgements Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through grant GSC 111 is gratefully acknowledged.  ... 
doi:10.1007/978-3-642-17884-9_19 fatcat:ydan2psyffc23othhziaotksxu

PERFORMANCE OF LIMITERS IN MODAL DISCONTINUOUS GALERKIN METHODS FOR 1-D EULER EQUATIONS
1-D 오일러 방정식에 관한 Modal 불연속 갤러킨 기법에서의 Limiter 성능 비교

A. Karchani, R.S. Myong
2016 Journal of computational fluids engineering  
Acknowledgments This work was supported by the National Research Foundation of Korea(NRF 2015-M1A3A3A02-010621), South Korea.  ...  The Rusanov(called as local Lax-Friedrichs) flux is applied for discretization of the inviscid flux functions at interfaces.  ...  In classical finite element methods(i.e., continuous finite element method), the global solution is discretized using the finite dimensional functions which are locally continuous in character with finite  ... 
doi:10.6112/kscfe.2016.21.2.001 fatcat:x7yobllvczhrzknkhpkzxvbrny
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