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.

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.

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I. Smears and E. Süli designed and analyzed a discontinuous Galerkin finite element method for the approximation of solutions to elliptic partial differential equations in nondivergence form. The results were proven, based on the assumption that the computational domain was convex and polytopal. In this paper, we extend this framework, allowing for Lipschitz continuous domains with piecewise curved boundaries.