论文标题
不连续的Galerkin的统一分析和$ C^0 $ - 汉密尔顿的罚款有限元方法 - 雅各比 - 贝尔曼和艾萨克斯方程
Unified analysis of discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations
论文作者
论文摘要
我们为一类不连续的盖尔金和$ c^0 $ -IP有限元近似的后验分析和先验误差界限，这些元素的二阶椭圆形汉密尔顿 - 雅各布 - 贝尔曼和艾萨克斯方程都具有cordes系数。我们证明了在有界凸域上带有电源系数的$ H^2 $的ISAACS方程中强解决方案的存在和独特性。然后，我们显示了两个和三个空间维度的简单网格上的分段多项式近似值的可计算基于剩余的误差估计器的可靠性和效率。我们引入了一个抽象框架，用于对广泛的数值方法的先验误差分析，并在Lipschitz连续性的三个关键条件下，离散的一致性和数值方法的强烈单调性证明了离散近似值的准选项。在这些条件下，我们还证明了最小规律性溶液中小型限制中数值近似值的收敛性。然后，我们证明该框架适用于文献中的一系列现有数值方法以及一些原始变体。我们结果的关键要素是对稳定项的原始分析。作为推论，我们还获得了离散的miranda--talenti不平等的概括，对分段多项式矢量场。
We provide a unified analysis of a posteriori and a priori error bounds for a broad class of discontinuous Galerkin and $C^0$-IP finite element approximations of fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions in $H^2$ of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for the a priori error analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda--Talenti inequality to piecewise polynomial vector fields.
