In this article, we propose and analyze an effective Hessian recovery strategy for the Lagrangian finite element method of arbitrary order. We prove that the proposed Hessian recovery method preserves polynomials of degree $k+1$ on general unstructured meshes and superconverges at a rate of $O(h^k)$ on mildly structured meshes. In addition, the method is proved to be ultraconvergent (two orders higher) for the translation invariant finite element space of any order. Numerical examples are presented to support our theoretical results.

This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization problem involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method. Deep learning, unfitted Nitsche method, interface problem, deep neural network. 78M10, 78A48, 47A70, 35P99 COPYRIGHT: © Global Science Press

In this article, we construct a $C^0$ linear finite element method for two fourth-order eigenvalue problems: the biharmonic and the transmission eigenvalue problems. The basic idea of our construction is to use gradient recovery operator to compute the higher-order derivatives of a $C^0$ piecewise linear function, which do not exist in the classical sense. For the biharmonic eigenvalue problem, the optimal convergence rates of eigenvalue/eigenfunction approximation are theoretically derived and numerically verified. For the transmission eigenvalue problem, the optimal convergence rate of the eigenvalues is verified by two numerical examples: one for constant refraction index and the other for variable refraction index. Compared with existing schemes in the literature, the proposed scheme is straightforward and simpler, and computationally less expensive to achieve the same order of accuracy.