A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

Kalchev, Delyan Z.; Manteuffel, Thomas A.

doi:10.1002/num.22480

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

Journal Article · Wed Jun 10 00:00:00 EDT 2020 · Numerical Methods for Partial Differential Equations

DOI:https://doi.org/10.1002/num.22480· OSTI ID:1592036

^[1]; Manteuffel, Thomas A. ^[2]

Univ. of Colorado, Boulder, CO (United States); Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Univ. of Colorado, Boulder, CO (United States)

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L² convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H^-1-type norm, but not L²-coercive. Nevertheless, the L² convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L² norm, or by properly choosing the finite element spaces, providing implicit control of the L² norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L² convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. Finally, we believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-07NA27344; NA0002376

OSTI ID:: 1592036

Alternate ID(s):: OSTI ID: 1632979

Report Number(s):: LLNL-JRNL--756408; 943871

Journal Information:: Numerical Methods for Partial Differential Equations, Journal Name: Numerical Methods for Partial Differential Equations Journal Issue: 6 Vol. 36; ISSN 0749-159X

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

References (27)

Weak solutions of nonlinear hyperbolic equations and their numerical computation Lax, Peter D. Communications on Pure and Applied Mathematics, Vol. 7, Issue 1 https://doi.org/10.1002/cpa.3160070112	journal	February 1954
High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation Anderson, R.; Dobrev, V.; Kolev, Tz. Journal of Computational Physics, Vol. 334 https://doi.org/10.1016/j.jcp.2016.12.031	journal	April 2017
Systems of conservation laws Lax, Peter; Wendroff, Burton Communications on Pure and Applied Mathematics, Vol. 13, Issue 2 https://doi.org/10.1002/cpa.3160130205	journal	May 1960
Curvilinear finite elements for Lagrangian hydrodynamics Dobrev, V. A.; Ellis, T. E.; Kolev, Tz. V. International Journal for Numerical Methods in Fluids, Vol. 65, Issue 11-12 https://doi.org/10.1002/fld.2366	journal	June 2010
Mixed (LL∗)−1 and LL∗ least-squares finite element methods with application to linear hyperbolic problems: Mixed (LL∗)−1 and LL∗ least-squares finite element methods with application to linear hyperbolic problems Kalchev, Delyan Z.; Manteuffel, Thomas A.; Münzenmaier, Steffen Numerical Linear Algebra with Applications, Vol. 25, Issue 3 https://doi.org/10.1002/nla.2150	journal	January 2018
Efficiency-basedh- andhp-refinement strategies for finite element methods De Sterck, H.; Manteuffel, T.; McCormick, S. Numerical Linear Algebra with Applications, Vol. 15, Issue 2-3 https://doi.org/10.1002/nla.567	journal	January 2008
Least-squares finite elements for first-order hyperbolic systems Carey, Graham F.; Jianng, B. N. International Journal for Numerical Methods in Engineering, Vol. 26, Issue 1 https://doi.org/10.1002/nme.1620260106	journal	January 1988
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
Numerical Methods for Conservation Laws LeVeque, Randall J. Lectures in Mathematics. ETH Zürich https://doi.org/10.1007/978-3-0348-8629-1	book	January 1992
Scalar conservation laws with discontinuous flux function: I. The viscous profile condition Diehl, S. Communications in Mathematical Physics, Vol. 176, Issue 1 https://doi.org/10.1007/BF02099361	journal	February 1996
SUPG finite element computation of compressible flows with the entropy and conservation variables formulations Le Beau, G. J.; Ray, S. E.; Aliabadi, S. K. Computer Methods in Applied Mechanics and Engineering, Vol. 104, Issue 3 https://doi.org/10.1016/0045-7825(93)90033-T	journal	May 1993
Implementation of the entropy viscosity method with the discontinuous Galerkin method Zingan, Valentin; Guermond, Jean-Luc; Morel, Jim Computer Methods in Applied Mechanics and Engineering, Vol. 253 https://doi.org/10.1016/j.cma.2012.08.018	journal	January 2013
Entropy viscosity method for nonlinear conservation laws Guermond, Jean-Luc; Pasquetti, Richard; Popov, Bojan Journal of Computational Physics, Vol. 230, Issue 11 https://doi.org/10.1016/j.jcp.2010.11.043	journal	May 2011
The Riemann problem of the Burgers equation with a discontinuous source term Fang, Beixiang; Tang, Pingfan; Wang, Ya-Guang Journal of Mathematical Analysis and Applications, Vol. 395, Issue 1 https://doi.org/10.1016/j.jmaa.2012.05.021	journal	November 2012
Finite Volume Methods for Hyperbolic Problems LeVeque, Randall J. https://doi.org/10.1017/CBO9780511791253	book	January 2002
A Comparative Study of Least-squares, SUPG and Galerkin Methods for Convection Problems Bochev, Pavel B.; Choi, Jungmin International Journal of Computational Fluid Dynamics, Vol. 15, Issue 2 https://doi.org/10.1080/10618560108970023	journal	October 2001
Why nonconservative schemes converge to wrong solutions: error analysis Hou, Thomas Y.; LeFloch, Philippe G. Mathematics of Computation, Vol. 62, Issue 206 https://doi.org/10.1090/S0025-5718-1994-1201068-0	journal	May 1994
Partial Differential Equations Evans, Lawrence Graduate Studies in Mathematics https://doi.org/10.1090/gsm/019	book	January 1998
Numerical Methods for Unconstrained Optimization and Nonlinear Equations Dennis, J. E.; Schnabel, Robert B. Classics in Applied Mathematics https://doi.org/10.1137/1.9781611971200	book	January 1996
Efficiency Based Adaptive Local Refinement for First-Order System Least-Squares Formulations Adler, J. H.; Manteuffel, T. A.; McCormick, S. F. SIAM Journal on Scientific Computing, Vol. 33, Issue 1 https://doi.org/10.1137/100786897	journal	January 2011
High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics Dobrev, Veselin A.; Kolev, Tzanio V.; Rieben, Robert N. SIAM Journal on Scientific Computing, Vol. 34, Issue 5 https://doi.org/10.1137/120864672	journal	January 2012
A Second-Order Maximum Principle Preserving Lagrange Finite Element Technique for Nonlinear Scalar Conservation Equations Guermond, Jean-Luc; Nazarov, Murtazo; Popov, Bojan SIAM Journal on Numerical Analysis, Vol. 52, Issue 4 https://doi.org/10.1137/130950240	journal	January 2014
The Auxiliary Space Preconditioner for the de Rham Complex Gopalakrishnan, J.; Neumüller, M.; Vassilevski, P. S. SIAM Journal on Numerical Analysis, Vol. 56, Issue 6 https://doi.org/10.1137/17M1153376	journal	January 2018
A First-Order System Least Squares Finite Element Method for the Shallow Water Equations Starke, Gerhard SIAM Journal on Numerical Analysis, Vol. 42, Issue 6 https://doi.org/10.1137/S0036142903438124	journal	January 2005
Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs De Sterck, H.; Manteuffel, Thomas A.; McCormick, Stephen F. SIAM Journal on Scientific Computing, Vol. 26, Issue 1 https://doi.org/10.1137/S106482750240858X	journal	January 2004
Numerical Conservation Properties of H (div)-Conforming Least-Squares Finite Element Methods for the Burgers Equation De Sterck, H.; Manteuffel, Thomas A.; McCormick, Stephen F. SIAM Journal on Scientific Computing, Vol. 26, Issue 5 https://doi.org/10.1137/S1064827503430758	journal	January 2005
Improved Least-squares Error Estimates for Scalar Hyperbolic Problems Bochev, P. B.; Choi, J. Computational Methods in Applied Mathematics, Vol. 1, Issue 2 https://doi.org/10.2478/cmam-2001-0008	journal	January 2001

Similar Records

Finite-difference approximations and entropy conditions for shocks

Journal Article · Wed Dec 31 23:00:00 EST 1975 · Commun. Pure Appl. Math.; (United States) · OSTI ID:7231693

Finite-difference approximations and entropy conditions for shocks

Technical Report · Wed Dec 31 23:00:00 EST 1975 · OSTI ID:7365921

Mixed and least‐squares finite element methods with application to linear hyperbolic problems

Journal Article · Thu Jan 18 19:00:00 EST 2018 · Numerical Linear Algebra with Applications · OSTI ID:1417505

Related Subjects

97 MATHEMATICS AND COMPUTING
Burgers equation
Helmholtz decomposition
conservation laws
finite element methods
hyperbolic balance laws
least-squares methods
negative-norm methods
space-time discretization
weak solutions

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

Citation Formats

References (27)

Similar Records

Related Subjects