A $C^0$ Linear Finite Element Method for a Second Order Elliptic Equation in Non-Divergence Form with Cordes Coefficients

TL;DR

This paper introduces a novel $C^0$ linear finite element method for second order elliptic equations in non-divergence form, utilizing gradient and Hessian recovery techniques to achieve accurate solutions with low degrees of freedom.

Contribution

The paper develops and rigorously analyzes a new $C^0$ linear FEM using gradient and Hessian recovery for non-divergence form elliptic equations, including applications to Monge-Ampère equations.

Findings

Proved unique solvability and $H^2$ error estimates for the method.

Achieved optimal error estimates in $L^2$ and $H^1$ norms for diagonal coefficients.

Observed superconvergence and applicability to curved domains.

Abstract

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-Amp\`{e}re equations.