## NUMERICAL CONSERVATION PROPERTIES OF H(div)-CONFORMING LEAST-SQUARES FINITE ELEMENT METHODS FOR THE BURGERS EQUATION ∗

Citations: | 1 - 0 self |

### BibTeX

@MISC{Sterck_numericalconservation,

author = {H. De Sterck and Thomas A. Manteuffel and Stephen F. Mccormick and Luke Olson},

title = {NUMERICAL CONSERVATION PROPERTIES OF H(div)-CONFORMING LEAST-SQUARES FINITE ELEMENT METHODS FOR THE BURGERS EQUATION ∗},

year = {}

}

### OpenURL

### Abstract

Abstract. Least-squares finite element methods (LSFEMs) for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector, or the associated flux potential, explicitly as additional dependent variables. This reformulation highlights the smoothness of the flux vector for weak solutions, namely, f(u) ∈ H(div, Ω). The standard least-squares (LS) finite element (FE) procedure is applied to the reformulated equations using H(div)-conforming FE spaces and a Gauss–Newton nonlinear solution technique. Numerical results are presented for the one-dimensional Burgers equation on adaptively refined space-time domains, indicating that the H(div)-conforming FE methods converge to the entropy weak solution of the conservation law. The H(div)-conforming LSFEMs do not satisfy a discrete exact conservation property in the sense of Lax and Wendroff. However, weak conservation theorems that are analogous to the Lax–Wendroff theorem for conservative finite difference methods are proved for the H(div)-conforming LSFEMs. These results illustrate that discrete exact conservation in the sense of Lax and Wendroff is not a necessary condition for numerical conservation but can be replaced by minimization in a suitable continuous norm. Key words. least-squares variational formulation, finite element discretization, Burgers equation, nonlinear hyperbolic conservation laws, weak solutions

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Citation Context ...ontinuities. It is well known that finite difference schemes that do not satisfy an exact discrete conservation property may converge to a function that is not a weak solution of the conservation law =-=[21, 22, 20, 26]-=-. In [21], Lax and Wendroff show that if finite difference schemes do satisfy an exact discrete conservation property and if they converge to a function boundedly almost everywhere (a.e.), then the fu... |

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Citation Context ... ψ would need to be replaced by a vector potential, which is the preimage of the divergence-free generalized flux vector, f(u), in the des1578 DE STERCK, MANTEUFFEL, MCCORMICK, AND OLSON Rham-diagram =-=[1, 4, 13]-=-. The de Rham-diagram is also instructive for the derivation of conforming vector FE spaces in higher dimensions [4]. Remark 2.9. Both reformulations (2.1) and (2.2) explicitly bring to the forefront ... |

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Citation Context ...atural, sharp a posteriori error estimator which can be used for efficient adaptive refinement. LSFEMs are widely used for elliptic PDEs [7] but have only recently been introduced for hyperbolic PDEs =-=[5, 6, 18, 14]-=-. They have not been applied to the H(div)-conforming reformulation we introduce in this paper, and the weak conservation properties of LSFEMs have not been analyzed theoretically. This paper is struc... |

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Citation Context ...atural, sharp a posteriori error estimator which can be used for efficient adaptive refinement. LSFEMs are widely used for elliptic PDEs [7] but have only recently been introduced for hyperbolic PDEs =-=[5, 6, 18, 14]-=-. They have not been applied to the H(div)-conforming reformulation we introduce in this paper, and the weak conservation properties of LSFEMs have not been analyzed theoretically. This paper is struc... |

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Citation Context ...atural, sharp a posteriori error estimator which can be used for efficient adaptive refinement. LSFEMs are widely used for elliptic PDEs [7] but have only recently been introduced for hyperbolic PDEs =-=[5, 6, 18, 14]-=-. They have not been applied to the H(div)-conforming reformulation we introduce in this paper, and the weak conservation properties of LSFEMs have not been analyzed theoretically. This paper is struc... |

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Citation Context ...) [21]. The FE convergence of our LSFEMs, namely, the L 2 convergence of u h to û, and the equivalence of our LS potential method to an H −1 formulation, will theoretically be analyzed elsewhere (see =-=[23]-=- for a discussion in the context of linear hyperbolic PDEs). Our H(div)-conforming LSFEM approach is clearly different from the numerical methods that are typically considered for nonlinear hyperbolic... |

1 |
least-squares error estimates for scalar hyperbolic problems
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Citation Context ...tural, sharp a posteriori error estimator which can be used for efficient adaptive refinement. LSFEMs are widely used for elliptic PDEs [7], but have only recently been introduced for hyperbolic PDEs =-=[5, 6, 18, 14]-=-. They have not been applied to the H(div)-conforming reformulation we introduce in this paper, and the weak conservation properties of LSFEMs have not been analyzed theoretically. This paper is struc... |