In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint ($\nabla \cdot \B = 0$) of its magnetic field $\B$. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements \cite{Li2005}, and another approach of the divergence cleaning technique given by Dedner et al. \cite{Dedner2002}. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

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For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.

Let σ be the geodesic inversion on a Heisenberg type group$N$with homogeneous dimension$Q$, and denote by$S$the jacobian of σ. We prove that, for $$ - \frac{1}{2}Q< \alpha< \frac{1}{2}Q$$ , the operators $$T_\alpha :f \mapsto S^{1/2 - \alpha /Q} (f \circ \sigma )$$ are bounded on certain homogeneous Sobolev spaces $$\mathcal{H}^\alpha (N)$$ if and only if$N$is an Iwasawa$N$-group.

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