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We provide a unified analysis of a posteriori and a priori error bounds for abroad class of discontinuous Galerkin and $C^0$-IP finite elementapproximations of fully nonlinear second-order ellipticHamilton--Jacobi--Bellman and Isaacs equations with Cordes coefficients. Weprove the existence and uniqueness of strong solutions in $H^2$ of Isaacsequations with Cordes coefficients posed on bounded convex domains. We thenshow the reliability and efficiency of computable residual-based errorestimators for piecewise polynomial approximations on simplicial meshes in twoand three space dimensions. We introduce an abstract framework for the a priorierror analysis of a broad family of numerical methods and prove thequasi-optimality of discrete approximations under three key conditions ofLipschitz continuity, discrete consistency and strong monotonicity of thenumerical method. Under these conditions, we also prove convergence of thenumerical approximations in the small-mesh limit for minimal regularitysolutions. We then show that the framework applies to a range of existingnumerical methods from the literature, as well as some original variants. A keyingredient of our results is an original analysis of the stabilization terms.As a corollary, we also obtain a generalization of the discreteMiranda--Talenti inequality to piecewise polynomial vector fields.

Unified analysis of discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations