In this paper, we study the superconvergence of the error for the discontinuous Galerkin (DG) finite element method for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the DG solution and the exact solution is ($k+2$)-th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is ($k+2$)-th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The superconvergence result in this paper leads to a new {\em a posteriori} error estimate. Our analysis is valid for arbitrary regular meshes and for $\mathcal{P}^k$ polynomials with arbitrary $k\geq1$, and for both periodic boundary conditions and for initial-boundary value problems. We perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal.

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We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

The public key of the Oil-Vinegar scheme consists of a set of m quadratic equations in m + n variables over a finite field F_q. Kipnis and Shamir broke the balanced Oil-Vinegar scheme where d = n − m = 0 by finding equivalent keys of the cryptosytem. Later their method was extended by Kipnis et al to attack the unbalanced case where 0 < d < m and d is small with a complexity of O(q^{d−1} m^4). This method uses the matrices associated with the quadratic polynomials in the public key, which needs to be symmetric and invertible. In this paper, we give an optimized search method for Kipnis el al’s attack. Moreover, for the case that the finite field is of characteristic 2, we find the situation becomes very subtle, which, however, was totally neglected in the original work of Kipnis et al. We show that the Kipnis-Shamir method does not work if the field characteristic is 2 and d is a small odd number, and we fix the situation by proposing an alternative method and give an equivalent key recovery attack of complexity O(q^{d+1} m^4). We also prove an important experimental observation by Ding et al for the Kipnis-Shamir attack on balanced Oil-Vinegar schemes in characteristic 2.

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