AI 中文总结
研究二维狄利克雷蒙日 - 安培方程光滑解，提出结合离散黑塞矩阵重构与局部投影的全离散格式，在特定条件下证明其局部收敛及最优阶收敛，通过数值实验验证收敛速率，此前该离散化方法未用于最小二乘分裂法。
AI 中文摘要
蒙日 - 安培方程的最小二乘分裂算法已成功应用于计算数年，但此类全离散分裂格式的收敛理论仍未可得。本文中，我们为二维狄利克雷蒙日 - 安培方程的光滑解引入并分析了一个有限元框架。所提出的格式将离散黑塞矩阵重构与行列式约束上的局部投影相结合。在离散米兰达 - 塔伦蒂估计和黑塞矩阵重构的标准逼近性质下，我们证明了迭代格式的局部收敛性及其极限在\(H^2\)型范数下到精确解的最优阶收敛性。我们验证了适用于协调\(C^1\)格式（包括阿吉里斯元）以及至少三次的\(C^0\)内部罚函数和间断伽辽金格式的估计；当\(|u|_{H^3(\Omega)}\)足够小时，二次\(C^0\)内部罚函数和间断伽辽金格式也涵盖在内。据我们所知，这些离散化方法此前尚未针对最小二乘分裂方法提出或分析过。数值实验证实了理论收敛速率。
英文摘要
The least-squares splitting algorithm for the Monge-Ampère equation has been used successfully in computations for several years, but a convergence theory for fully discrete splitting schemes of this type has remained unavailable. In this work, we introduce and analyze a finite element framework for smooth solutions of the Dirichlet Monge-Ampère equation in two dimensions. The proposed schemes combine a discrete Hessian reconstruction with a local projection onto the determinant constraint. Under a discrete Miranda-Talenti estimate and standard approximation properties of the Hessian reconstruction, we prove local convergence of the iterative scheme and optimal-order convergence of its limit to the exact solution in an $H^2$-type norm. We verify the estimates for conforming $C^1$ schemes, including the Argyris element, and for $C^0$-interior penalty and DG schemes of degree at least three; quadratic $C^0$-interior penalty and DG schemes are also covered when $|u|_{H^3(Ω)}$ is sufficiently small. To the best of our knowledge, these discretizations have not previously been proposed or analyzed for least-squares splitting methods. Numerical experiments confirm the theoretical convergence rates.
Comments25 pages
