论文标题
连续有限元素满足强大的离散米兰达（Talenti）身份
Continuous finite elements satisfying a strong discrete Miranda--Talenti identity
论文作者
论文摘要
本文介绍了连续的$ h^2 $ - 共有的有限元素在两个和三个空间尺寸中，从而满足了强大的离散米兰达（TALENTI）的不平等，从某种意义上说，分段Hessian的全球$ l^2 $规范受$ l^2 $ laplacian的$ l^2 $规范。该结构基于在顶点（2D）或边缘（3D）上具有$ C^1 $连续性的全球连续有限元函数。作为一种应用，这些有限元素用于在有线条件下以非差异形式近似椭圆形的方程，而没有其他稳定项。对于三个维度的Biharmonic方程，所提出的方法比相同顺序的现有不合格方案具有较小的自由度。两个维度和三个维度的数值结果证实了所提出的方案的实际可行性。
This article introduces continuous $H^2$-nonconforming finite elements in two and three space dimensions which satisfy a strong discrete Miranda--Talenti inequality in the sense that the global $L^2$ norm of the piecewise Hessian is bounded by the $L^2$ norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with $C^1$ continuity on the vertices (2D) or edges (3D). As an application, these finite elements are used to approximate uniformly elliptic equations in non-divergence form under the Cordes condition without additional stabilization terms. For the biharmonic equation in three dimensions, the proposed methods has less degrees of freedom than existing nonconforming schemes of the same order. Numerical results in two and three dimensions confirm the practical feasibility of the proposed schemes.
