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 Poincar\'e 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.
翻译:我们证明了适应性不连续的Galerkin 和 $C$0 内置惩罚方法对完全非线性第二序列的埃密尔顿-Jacobi-Bellman和Isaacs等式与Cordes系数的趋同。我们考虑在两个和三个空间维度上采用一系列广泛的适应性改进符合简化的短片方法,固定但任意的多元度大于或等于两个空间维度。我们方法的一个关键要素是对限制空间的一种新颖的内在定性,它使我们能够查明不兼容的有限元素功能的约束序列的薄弱限度。我们为限制空间和一些原始的辅助功能空间提供了详细的理论,这对于适应性不兼容方法解决更一般性的问题(包括Poincar\'e和痕量不平等)具有独立的利益,证明功能的密度,只在有限骨架的众多面上出现非加速跳跃,用有限元素函数的近似结果和微弱的融合结果。