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 the 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.

我们证明了自适应不连续Galerkin和\（C ^ 0 \）的收敛性非线性的具有Cordes系数的二阶椭圆形Hamilton-Jacobi-Bellman和Isaacs方程的内部惩罚方法。我们考虑了在二维和三个空间维上自适应精炼的符合简单网格的广泛方法，其中固定但任意多项式的阶数都大于或等于2。我们方法的关键要素是极限空间的新颖内在表征，它使我们能够识别不合格有限元函数的有界序列的弱极限。我们为极限空间以及一些原始的辅助函数空间提供了详细的理论，这对于适应更不协调的问题（包括庞加莱和迹线不等式）的自适应不符合方法具有独立的意义，