论文标题
适应性不连续的盖尔金和$ C^0 $ - 汉密尔顿的有限元方法的融合 - 雅各比 - 贝尔曼和艾萨克斯方程
Convergence of adaptive 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
我们证明了自适应不连续的Galerkin和$ C^0 $ C^0 $ CINDION罚款方法，用于完全非线性的二阶椭圆形汉密尔顿 - Jacobi-Bellman和Isaacs方程，具有Cordes系数。我们考虑了在两个和三个空间维度的适应性精制构造的简单网格上的广泛方法，其固定但任意的多项式度大于或等于两个。我们方法的关键要素是对极限空间的新型内在表征，使我们能够确定不合格有限元函数的有界序列的弱极限。我们为限制空间提供了详细的理论，以及一些原始的辅助功能空间，这是针对更普遍的问题的自适应不合格方法的独立兴趣，包括Poincaré和Trace不平等，功能的密度证明，不断变化的跳跃证明，仅在极限骨骼的许多面上有限多个面孔，均通过有限的元素功能和弱元素函数结果，并获得了cornervections cornerviens cornervensions的差异结果。
We prove the convergence of adaptive discontinuous Galerkin and $C^0$-interior penalty methods for fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman and Isaacs equations with Cordes coefficients. We consider a broad family of methods on adaptively refined conforming simplicial meshes in two and three space dimensions, with fixed but arbitrary polynomial degrees greater than or equal to two. A key ingredient of our approach is a novel intrinsic characterization of the limit space that enables us to identify the weak limits of bounded sequences of nonconforming finite element functions. We provide a detailed theory for the limit space, and also some original auxiliary functions spaces, that is of independent interest to adaptive nonconforming methods for more general problems, including Poincaré and trace inequalities, a proof of density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, approximation results by finite element functions and weak convergence results.
