We design and analyze a \(C^0\) interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions.

中文翻译：

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凸多边形域上 Monge–Ampère 方程的凸性执行 $${C}^{{0}}$$ C 0 内部惩罚方法

我们设计并分析了一个\(C^0\)内部惩罚方法，用于逼近凸多边形域上 Monge-Ampère 方程的 Dirichlet 边值问题的经典解。该方法基于增强型三次拉格朗日有限元，该有限元能够增强近似解的凸性。给出了证实先验和后验误差估计的数值结果。从数值实验中还观察到，该方法可以捕获某些弱解。