论文标题
A $ C^0 $线性有限元法，用于二阶椭圆方程的非差异形式，带有电源系数
A $C^0$ Linear Finite Element Method for a Second Order Elliptic Equation in Non-Divergence Form with Cordes Coefficients
论文作者
论文摘要
在本文中，我们开发了一种基于梯度恢复的线性（GRBL）有限元方法（FEM）和以非差异形式的二阶椭圆方程的基于Hessian恢复的线性（HRBL）FEM。将椭圆方程式施加到对称的非发散弱公式中，其中涉及未知函数的二阶导数。我们使用梯度和Hessian恢复操作员来计算线性有限元近似值的二阶导数。尽管由于线性元素的自由度低（DOF），因此提出的方案的实施非常容易且直接，但这些方法的性能具有竞争力。严格证明了GRBL方案的唯一解决性和$ H^2 $ eminorm误差估计。当系数为对角线时，已证明了$ l^2 $ norm和$ h^1 $ eminorm的最佳错误估计，这已通过数值实验证实。还观察到了误差的超授权。此外，我们的方法可以处理具有弯曲边界的计算域，而不会因边界近似而丧失准确性。最后，已成功应用了所提出的数值方法来求解完全非线性的蒙格 - 安培方程。
In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and a Hessian recovery based linear (HRBL) FEM for second order elliptic equations in non-divergence form. The elliptic equation is casted into a symmetric non-divergence weak formulation, in which second order derivatives of the unknown function are involved. We use gradient and Hessian recovery operators to calculate the second order derivatives of linear finite element approximations. Although, thanks to low degrees of freedom (DOF) of linear elements, the implementation of the proposed schemes is easy and straightforward, the performances of the methods are competitive. The unique solvability and the $H^2$ seminorm error estimate of the GRBL scheme are rigorously proved. Optimal error estimates in both the $L^2$ norm and the $H^1$ seminorm have been proved when the coefficient is diagonal, which have been confirmed by numerical experiments. Superconvergence in errors has also been observed. Moreover, our methods can handle computational domains with curved boundaries without loss of accuracy from approximation of boundaries. Finally, the proposed numerical methods have been successfully applied to solve fully nonlinear Monge-Ampère equations.
