Unified analysis of discontinuous Galerkin andC⁰-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations

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.

Mixed Finite Element Approximation of Periodic Hamilton--Jacobi--Bellman Problems With Application to Numerical Homogenization

“…Future work will focus on the numerical homogenization of other fully nonlinear partial differential equations such as the Isaacs equation. The strong H 2 solution of Isaacs equations with Cordes coefficients has recently been discussed in [27] and can be used as a framework to study its numerical homogenization.…”

Convergence of Adaptive Discontinuous Galerkin and $$C^0$$-Interior Penalty Finite Element Methods for Hamilton–Jacobi–Bellman and Isaacs Equations

“…Note that in the context of these problems, DG and C 0 -IP methods are examples of nonconforming methods, since the appropriate functional setting is in H 2 -type spaces. In [36], we provide a unified analysis of a posteriori and a priori error bounds for a wide family of DG and C 0 -IP methods, where we also show that the original method of [49,50], along with many related variants, is quasi-optimal in the sense of near-best approximations without any additional regularity assumptions, along with convergence in the small mesh-limit for minimal regularity solutions.…”

“…In order to focus on the most important aspects of analysis, we shall restrict our attention to Isaacs and HJB equations without lower-order terms, although we note that the approach we consider here easily accommodates problems with lower-order terms, see [36,50,51]. More precisely, let the real valued functions a i j = a ji and f belong to C( × A × B) for each i, j = 1, .…”

“…In Sect. 3, we state our main assumptions on the problem (1.1) and recall some well-posedness results from [36,50]. Section 3 then introduces the family of adaptive DG and C 0 -IP methods that are considered and states our main result on convergence of the adaptive algorithm in Theorem 3.…”

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

“…An alternative approach, recently proposed in [35] in the context of nonlinear PDEs in nondivergence form, is to reconstruct the solution into C 1 -conforming spaces introduced in [17,45]. While this allows us to avoid problems with element geometries, it introduces the disadvantage in the current context that the resulting error estimate would gain an additional suboptimality of order p d in d spatial dimensions, due to the repeated application of a polynomial inverse estimate apparently necessary for the analysis.…”