带Cordes系数的HJB方程的虚单元方法
Virtual Element Methods for HJB Equations with Cordes Coefficients
AI总结：
针对带Cordes系数的二阶椭圆型HJB方程，提出协调与非协调虚单元方法，通过稳定化项建立适定性，推导离散$H^2$范数最优误差估计，用半光滑牛顿法线性化，数值实验验证了方法有效性与理论结果。
摘要：
本文提出并分析了用于求解带Cordes系数的完全非线性二阶椭圆型Hamilton-Jacobi-Bellman（HJB，哈密顿-雅可比-贝尔曼）方程的协调与非协调虚单元方法（VEMs）。通过引入稳定化项，我们建立了所提方法的适定性，从而无需构造离散Miranda-Talenti估计。我们推导了两种数值格式在离散$H^2$范数下的最优误差估计。此外，采用半光滑牛顿法对离散问题进行线性化。本文还提供了若干使用最低阶VEMs的数值实验，以证明所提方法的有效性并验证我们的理论结果。
英文摘要：
In this paper, we propose and analyze both conforming and nonconforming virtual element methods (VEMs) for the fully nonlinear second-order elliptic Hamilton-Jacobi-Bellman (HJB) equations with Cordes coefficients. By incorporating stabilization terms, we establish the well-posedness of the proposed methods, thus avoiding the need to construct a discrete Miranda-Talenti estimate. We derive the optimal error estimate in the discrete $H^2$ norm for both numerical formulations. Furthermore, a semismooth Newton's method is employed to linearize the discrete problems. Several numerical experiments using the lowest-order VEMs are provided to demonstrate the efficacy of the proposed methods and to validate our theoretical results.
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