Finite Element Approximation of Hamilton-Jacobi-Bellman equations with nonlinear mixed boundary conditions

TL;DR

This paper develops and analyzes a finite element method for solving Hamilton-Jacobi-Bellman equations with complex boundary conditions, demonstrating strong convergence and stability on unstructured meshes.

Contribution

It introduces a monotone P1 finite element approach for HJB equations with mixed boundary conditions, including discontinuities and degeneracies, ensuring convergence to viscosity solutions.

Findings

Proves uniform convergence of the method.

Handles discontinuous boundary operators.

Uses IMEX schemes for time discretization.

Abstract

We show strong uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate diffusions. Boundary operators can generally be discontinuous across face-boundaries and type changes. Robin-type boundary conditions are discretised via a lower Dini derivative. In time the Bellman equation is approximated through IMEX schemes. Existence and uniqueness of numerical solutions follows through Howard's algorithm. Keywords: Finite element method, Hamilton-Jacobi-Bellman equation, Mixed boundary conditions, Fully nonlinear equation, Viscosity solution