  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.
