Numerical Conservation Properties of H(div)-Conforming Least-Squares Finite Element Methods for the Burgers Equation

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, $\bff(u) \in H({\rm div},\Omega)$. 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.

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