This paper is concerned with C⁰ (non-Lagrange) finite element approximations of the linear elliptic equations in non-divergence form and the Hamilton–Jacobi–Bellman (HJB) equations with Cordes coefficients. Motivated by the Miranda–Talenti estimate, a discrete analog is proved once the finite element space is C⁰ on the \((n-1)\)-dimensional subsimplex (face) and \(C^1\) on \((n-2)\)-dimensional subsimplex. The main novelty of the non-standard finite element methods is to introduce an interior stabilization term to argument the PDE-induced variational form of the linear elliptic equations in non-divergence form or the HJB equations. As a distinctive feature of the proposed methods, no stabilization parameter is involved in the variational forms. As a consequence, the coercivity constant (resp. monotonicity constant) for the linear elliptic equations in non-divergence form (resp. the HJB equations) at discrete level is exactly the same as that from PDE theory. The quasi-optimal order error estimates as well as the convergence of the semismooth Newton method are established. Numerical experiments are provided to validate the convergence theory and to illustrate the accuracy and computational efficiency of the proposed methods.

本文涉及非散度形式的线性椭圆方程和具有Cordes系数的Hamilton–Jacobi–Bellman（HJB）方程的C ⁰（非Lagrange）有限元逼近。由米兰达-Talenti估计动机，一离散模拟证明一旦有限元空间是Ç ⁰上\（第（n-1）\）维subsimplex（面）和\（C ^ 1 \）上\（（ n-2）\）维次简单体。非标准有限元方法的主要新颖之处在于引入了一个内部稳定项，以非散度形式或HJB方程对PDE诱导的线性椭圆方程的变分形式进行论证。作为所提出方法的显着特征，变型形式不涉及稳定参数。结果，离散级的非散度形式的线性椭圆方程（分别为HJB方程）的矫顽力常数（分别为单调性常数）与PDE理论中的完全相同。建立了准最优阶误差估计以及半光滑牛顿法的收敛性。提供数值实验以验证收敛理论并说明所提出方法的准确性和计算效率。