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

TL;DR

This paper provides a unified theoretical framework for analyzing discontinuous Galerkin and $C^0$-IP finite element methods applied to complex fully nonlinear second-order elliptic equations, ensuring error bounds, convergence, and applicability to various methods.

Contribution

It introduces an abstract framework for a priori error analysis, proving quasi-optimality and convergence for a broad class of numerical methods solving Hamilton--Jacobi--Bellman and Isaacs equations.

Findings

Proved existence and uniqueness of strong solutions in $H^2$ for Isaacs equations.

Established reliability and efficiency of residual-based error estimators.

Demonstrated the framework's applicability to existing and new numerical methods.

Abstract

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.