A simple finite element method for linear hyperbolic problems

Mu, Lin; Ye, Xiu

doi:10.1016/j.cam.2017.08.025

Title: A simple finite element method for linear hyperbolic problems

Abstract

Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.

Authors:

; Ye, Xiu

Publication Date:: Thu Mar 01 00:00:00 EST 2018

Research Org.:: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE; National Science Foundation (NSF)

OSTI Identifier:: 1846816

Alternate Identifier(s):: OSTI ID: 1407723; OSTI ID: 1495741

Grant/Contract Number:: ERKJE45; AC05-00OR22725; DMS-1620016

Resource Type:: Published Article

Journal Name:: Journal of Computational and Applied Mathematics

Additional Journal Information:: Journal Name: Journal of Computational and Applied Mathematics Journal Volume: 330 Journal Issue: C; Journal ID: ISSN 0377-0427

Publisher:: Elsevier

Country of Publication:: Belgium

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Finite element methods; Hyperbolic equations

Citation Formats


                    Mu, Lin, and Ye, Xiu. A simple finite element method for linear hyperbolic problems.  Belgium: N. p., 2018. 
Web.  doi:10.1016/j.cam.2017.08.025.

Copy to clipboard


                    Mu, Lin, & Ye, Xiu. A simple finite element method for linear hyperbolic problems.  Belgium.  https://doi.org/10.1016/j.cam.2017.08.025

Copy to clipboard


                    Mu, Lin, and Ye, Xiu. Thu .  
"A simple finite element method for linear hyperbolic problems".  Belgium.  https://doi.org/10.1016/j.cam.2017.08.025.

Copy to clipboard


                    
@article{osti_1846816,

  title        = {A simple finite element method for linear hyperbolic problems},

  author       = {Mu, Lin and Ye, Xiu},

  abstractNote = {Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.},

  doi          = {10.1016/j.cam.2017.08.025},

  journal      = {Journal of Computational and Applied Mathematics},

  number       = C,

  volume       = 330,

  place        = {Belgium},

  year         = {Thu Mar 01 00:00:00 EST 2018},

  month        = {Thu Mar 01 00:00:00 EST 2018}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Publisher's Version of Record
https://doi.org/10.1016/j.cam.2017.08.025

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 8 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations
journal, August 2008

Burman, Erik; Quarteroni, Alfio; Stamm, Benjamin
Journal of Scientific Computing, Vol. 43, Issue 3
DOI: 10.1007/s10915-008-9232-6

Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems
journal, September 2007

Burman, E.; Stamm, B.
Journal of Scientific Computing, Vol. 33, Issue 2
DOI: 10.1007/s10915-007-9149-5

Finite element methods for linear hyperbolic problems
journal, September 1984

Johnson, Claes; Nävert, Uno; Pitkäranta, Juhani
Computer Methods in Applied Mechanics and Engineering, Vol. 45, Issue 1-3
DOI: 10.1016/0045-7825(84)90158-0

Improved Least-squares Error Estimates for Scalar Hyperbolic Problems
journal, January 2001

Bochev, P. B.; Choi, J.
Computational Methods in Applied Mathematics, Vol. 1, Issue 2
DOI: 10.2478/cmam-2001-0008

Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs
journal, January 2004

De Sterck, H.; Manteuffel, Thomas A.; McCormick, Stephen F.
SIAM Journal on Scientific Computing, Vol. 26, Issue 1
DOI: 10.1137/S106482750240858X

Stabilized hp -Finite Element Methods for First-Order Hyperbolic Problems
journal, January 2000

Houston, Paul; Schwab, Christoph; Süli, Endre
SIAM Journal on Numerical Analysis, Vol. 37, Issue 5
DOI: 10.1137/S0036142998348777

Discontinuous Galerkin Methods for First-Order Hyperbolic Problems
journal, December 2004

Brezzi, F.; Marini, L. D.; SÜLi, E.
Mathematical Models and Methods in Applied Sciences, Vol. 14, Issue 12
DOI: 10.1142/S0218202504003866

An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
journal, January 1986

Johnson, C.; Pitk{äranta, J.
Mathematics of Computation, Vol. 46, Issue 173
DOI: 10.1090/S0025-5718-1986-0815828-4

A weak Galerkin mixed finite element method for second order elliptic problems
journal, May 2014

Wang, Junping; Ye, Xiu
Mathematics of Computation, Vol. 83, Issue 289
DOI: 10.1090/S0025-5718-2014-02852-4

Similar Records in DOE PAGES and OSTI.GOV collections:

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

Journal Article Mu, Lin ; Wang, Junping ; Ye, Xiu - SIAM Journal on Scientific Computing

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for bothmore »« less
Cited by 15
https://doi.org/10.1137/16M1083244

Full Text Available
A hybridized formulation for the weak Galerkin mixed finite element method

Journal Article Mu, Lin ; Wang, Junping ; Ye, Xiu - Journal of Computational and Applied Mathematics

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising frommore »« less
Cited by 15
https://doi.org/10.1016/j.cam.2016.01.004
An a posteriori error estimator for the weak Galerkin least-squares finite-element method

Journal Article Adler, James H. ; Hu, Xiaozhe ; Mu, Lin ; ... - Journal of Computational and Applied Mathematics

In this paper, we derive an a posteriori error estimator for the weak Galerkin least-squares (WG-LS) method applied to the reaction–diffusion equation. Here, we show that this estimator is both reliable and efficient, allowing it to be used for adaptive refinement. Due to the flexibility of the WG-LS discretization, we are able to design a simple and straightforward refinement scheme that is applicable to any shape regular polygonal mesh. Finally, we present numerical experiments that confirm the effectiveness of the estimator, and demonstrate the robustness and efficiency of the proposed adaptive WG-LS approach.
Cited by 7
https://doi.org/10.1016/j.cam.2018.09.049
A new weak Galerkin finite element method for elliptic interface problems

Journal Article Mu, Lin ; Wang, Junping ; Ye, Xiu ; ... - Journal of Computational Physics

We introduce and analyze a new weak Galerkin (WG) finite element method in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H1 and L2 norms are established for the present WG finite element solutions. We conducted extensive numerical experiments inmore »« less
Cited by 76
https://doi.org/10.1016/j.jcp.2016.08.024

Full Text Available
A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

Journal Article Bailey, Teresa S. ; Adams, Marvin L. ; Yang, Brian ; ... - Journal of Computational Physics

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer's vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer's. However, since the PWL method producesmore »« less
https://doi.org/10.1016/j.jcp.2007.11.026

Similar Records

Title: A simple finite element method for linear hyperbolic problems

Abstract

Citation Formats

Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations journal, August 2008

Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems journal, September 2007

Finite element methods for linear hyperbolic problems journal, September 1984

Improved Least-squares Error Estimates for Scalar Hyperbolic Problems journal, January 2001

Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs journal, January 2004

Stabilized hp -Finite Element Methods for First-Order Hyperbolic Problems journal, January 2000

Discontinuous Galerkin Methods for First-Order Hyperbolic Problems journal, December 2004

An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation journal, January 1986

A weak Galerkin mixed finite element method for second order elliptic problems journal, May 2014

Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations
journal, August 2008

Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems
journal, September 2007

Finite element methods for linear hyperbolic problems
journal, September 1984

Improved Least-squares Error Estimates for Scalar Hyperbolic Problems
journal, January 2001

Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs
journal, January 2004

Stabilized hp -Finite Element Methods for First-Order Hyperbolic Problems
journal, January 2000

Discontinuous Galerkin Methods for First-Order Hyperbolic Problems
journal, December 2004

An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
journal, January 1986

A weak Galerkin mixed finite element method for second order elliptic problems
journal, May 2014