Virtual Element Methods for HJB Equations with Cordes Coefficients

TL;DR

This work develops and analyzes both $C^1$-conforming and $C^0$-nonconforming virtual element methods for fully nonlinear Hamilton–Jacobi–Bellman equations with Cordes coefficients on polygonal domains. By incorporating a stabilization term, the authors ensure well-posed discrete problems without relying on a discrete Miranda–Talenti estimate and prove optimal a priori error bounds in the discrete $H^2$-norm, while linearizing the schemes via a semismooth Newton method. Theoretical results are complemented by extensive numerical experiments on diverse meshes and problems, confirming the predicted convergence rates for the $H^2$-seminorm and the $L^2$- and $H^1$-type errors for the solution and its derivatives. The approach offers robust, high-fidelity discretizations on general meshes for HJB equations in nondivergence form, with potential extensions to related fully nonlinear PDEs in stochastic control and finance.

Abstract

In this paper, we propose and analyze both conforming and nonconforming virtual element methods (VEMs) for the fully nonlinear second-order elliptic Hamilton-Jacobi-Bellman (HJB) equations with Cordes coefficients. By incorporating stabilization terms, we establish the well-posedness of the proposed methods, thus avoiding the need to construct a discrete Miranda-Talenti estimate. We derive the optimal error estimate in the discrete $H^2$ norm for both numerical formulations. Furthermore, a semismooth Newton's method is employed to linearize the discrete problems. Several numerical experiments using the lowest-order VEMs are provided to demonstrate the efficacy of the proposed methods and to validate our theoretical results.

Figures (11)

• Figure 1: The degrees of freedom with $k=2$ (left) and $k=3$ (right)
• Figure 1: The degrees of freedom with $k=2$ (left) and $k=3$ (right).
• Figure 1: Meshes of $\mathcal{K}^1$ (left) and $\mathcal{K}^2$ (right)
• Figure 2: Errors for $C^1$-conforming (left) and $C^0$-nonconforming (right) VEM on the mesh $\mathcal{K}^2$
• Figure 3: Meshes of $\mathcal{K}^3$ (left), $\mathcal{K}^4$ (middle) and $\mathcal{K}^5$ (right)

Theorems & Definitions (25)

• Lemma 1.1
• Definition 2.1
• Theorem 2.2
• Lemma 3.3
• Lemma 3.4
• Proof 1
• Remark 4.1
• Lemma 4.2
• Proof 2
• Lemma 4.3