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I'm used to displacement forumlation of elastic wave equation: $$ \nabla \cdot \sigma (u) + F = \rho \ddot{ u } $$ where $u$ is the primary variable. Recenty I started experimenting with DG and in almost every paper the stress-velocity formulation is used as a "conservative or divergence form". What's special about this formulation? I can't figure out why they use Inerior Penalty with displacement equation and other fluxes like Lax-Friedrich's with a system of $v, \sigma$. I would be thankful for any kind of paper, book or resource.

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The stress-velocity formulation has been used extensively in DG context on account of the fact that it can lead to a first order system form of the elastic equation. The latter proved to lend itself more easily to a DG formulation.

However, recently a DG formulation of the elastic equation in the second-order form has been proposed, which you may find useful, see here.

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