Work

Virtual element methods for HJB equations with Cordes coefficients

Year: 2025

Type: article

Abstract: • Develop C 1 -conforming and C 0 -nonconforming virtual element methods for Hamilton-Jacobi-Bellman equations in nondivergence form under the Cordes condition. • Introduce a novel stabilization strategy that ensures strong monotonicity and stability without discrete Miranda-Talenti identities. • Establish well-posedness and a priori error estimates with optimal convergence in mesh-broken norms. • The methods are polygonal-mesh flexible, efficiently implementable, and well-suited for low-regularity fully nonlinear PDEs. 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.

Source: Computer Methods in Applied Mechanics and Engineering

DOI: 10.1016/j.cma.2025.118362

Authors: Ying Cai, Hailong Guo, Zhimin Zhang

Institutions: University of Science and Technology of China, Beijing Computational Science Research Center, The University of Melbourne, Wayne State University

Language: English

Topic: Advanced Numerical Methods in Computational Mathematics

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,830

Funders: University of Melbourne, National Natural Science Foundation of China