This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge-Amp\`ere equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.
翻译:本文建议通过统一的椭圆形汉密尔顿-Jacobi-Bellman等式,将蒙-安普-安普-埃雷方程式在平面方程式域中的蒙-安普-安普-安普-埃雷方程式正规化。规范化的问题有一个独特的强有力的解决方案,用有限的元素提供美元,可供离散的元素使用。这项工作使当地统一地将乌瓦雷普西隆元与亚历山大-安普-安普-安普-安普-安普-安普-安普-平面的方程式趋同为美元,作为规范化参数的美元接近值接近值为0美元。提出了一种混合的拉瓦雷普西隆近值的有限元素方法,并显示常规化的限定要素方案是当地统一一致的。数字实验为单项解决方案的有效接近提供了经验证据。